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» MATHEMATICS 



PRACTICAL MEN: 

BEnre 

A COMMON-PLACE BOOK 

or 

PURE AND MIXED MATHEMATICS, 

DBSIOVSD OHHILT rOR THB VSE OF 

CIVIL ENGINEERS, ARCHITECTS, AND SURVEYORS. 



BY OLINTHUS GBEGOBY, LL.D., F.R.A.S. 



THIRD EDITION, REVISED AND ENLARGED 
BY HENRY LAW, 

cnriL xiranriVB. 



LONDON: 
JOHN WEALE, 69, HIGH HOLBORN. 

1848. 



LONDON : 
OBOROB WOODFALL AND SOX, 

AJfOCL COURT. 8KINNBR STRBKT. 




TU 

SIR ISAMBART BRUNEL, F.RS., 

na BO. xra 
THIS WORK. 

DESIGNED FOB THE USE OF MEMBEBS OF THAT FBOFESSION WHICH 
HE HAS 80 OBEATLY ADVANCED, 

IS INSCBIBED, 

, IN GBATEFUL ACKKOWLEDOMENT OF MANY KINDNESSES, 

BY HIS OBUOED FBIEND AND PUPIL, 

HENRY LAW. 



AUT HOE'S PREFACE. 



Ths wot\l now presented to the public bad its origin in a desire 
which I felt to draw up an Essay on the principles and apph'cations 
of the Tnechanical sciences for the use of the younger members of the 
InsdtutioQ of Civil Engineers. The eminent individuals who are 
deserredly r^arded as the main pillars of that useful Institution, 
stand in need of no such instructions as are in my power to impart : 
kt it seemed expedient to prepare an Essay, comprised within 
moderate limits, which might furnish scientific instruction for the 
many young men of ardour and enterprise who have of late years 
deToted themseWes to the interesting and important profession, of 
whose members that InsUtution is principally constituted. My first 
design was to compose a paper which might be read at one or two 
of the meetings of that Society ; but, as often happens in such cases, 
the embryo thought has grown, during meditation, from an essay to a 
book ; and what was first meant to be a very compendious selection 
of principles and rules, has, in its execution, assumed the appearance 
of a systematic aualysis of principles, theorems, rules, and tables. 

Indeed, tbe circumstances in which the inhabitants of this country 
are now placed, with regard to the Ioto and acquisition of knowledge 
impelled me, almost unconscionsly, to such an extension of my ori- 
ginal plan, as sprung from a desire to contribute to the instruction of 
that numerous class, the practical mechanics of this country. Besides 
the early disadvantages under which many of them have laboured, 
there is another which results from the activity of their pursuits. 
Unable, therefore, to go through the details of an extensive systematie 
course, they must, for the most part, be satisfied with imperfect views 
of theories and principles, and take much upon trust : an evil, how- 
ever, which tbe establishment of Societies, and the composition of 
treatises, with an express view to their benefit, will probably soon 
diminish. 

Lord Brouorjim, in his ^^ PracHeal Observations upon the Educa^ 
turn of the Peofk^ remarks that " a most essential service will be 



▼1 AUTHORS PREFACE. 

rendered to the cause of knowle<lge, by him who shall devote his 
time to the composition of elementary treatises on the Mathematics, 
sufficiently clear, and yet sufficiently compendious, to exemplify the 
metliod of reasoning employed in that science, and to impart an 
accurate knowledge of the most useful fundamental propositions, 
with their application to practical purposes; and treatises upon Na- 
tural Philosophy, which may teach the great principles of physics, 
and their practical application, to readers who have but a general 
knowledge of Mathematics, or who are even wholly ignorant of the 
science beyond the common rules of arithmetic/' And again, '^ He 
who shall prepare a treatise simply and concisely imfolding the doc- 
trines of Algebra, Geometry, and Mechanics, and adding examples 
calculated to strike the imagination, of their connection with other 
branches of knowledge, and with the arts of common life, may fairly 
claim a large share in that rich harvest of discovery and invention 
which must be reaped by the thousands of ingenious and active men, 
thus enabled to bend their faculties towards objects at once useful 
and sublime." 

I do not attempt to persuade myself that the present volume will 
be thought adequately to supply the desiderata to which these pas- 
sages advert : yet I could not but be gratified, after full two-thirds 
of it were written, to find that the views which guided me in its 
execution accorded so far with the judgment of an individual, dis- 
tinguished, as Lord Brougham was, in early life, for the elegance 
and profundity of his mathematical researches. 

With a view to the elementary instruction of those who have not 
previously studied Mathematics, I have commenced with brief, but, I 
hope, perspicuous, treatises on Arithmetic and Algebra; a competent 
acquaintance with both of these being necessary to ensure that ac- 
curacy in computation which every practical man ought to attain, 
and that ready comprehension of scientific theorems and formul» 
which becomes the key to the stores of higher knowledge. As no 
man sharpens his tool or his weapon, merely that it may be sharp, 
but that it may be the fitter for use ; so no thoughtful man learns 
Arithmetic and Algebra for the mere sake of knowing those branches 
of science, but that he may employ them ; and these being possessed 
as valuable pre-requisites, the course of an author is thereby facili- 
tated : for then, while he endeavours to express even common mat- 
ters so that the learned shall not be disgusted, he may so express the 
more abstract and difficult that the comparatively ignorant (and the 



▲UTHOBS PRBPACI. yil 

DereliQowledge of Jirithmetie tmdAl^bra is, in our times, compara^ 
the i^orence,') may practically understand and apply them. 
Mter the first 97 I>age8, the remaining matter is synoptical. The 
leoend topics of Oeomctry, Trigonometry, Conic Sections, Curves, 
Pmpectiye, Mensuration, Statics, Dynamics, Hydrostatics, Hydro- 
dyiuumcs, and Pneumatics are thus treated. The definitions and 
pnudples are exhibited in an orderly series ; but investigations and 
demonstiations are only sparingly introduced. This portion of the 
voik is aldn in its nature to a syllabus of a Course of Lectures on 
tiie deportments of science which it treats; with this difference, 
however, occasioned by tbe leading object of the publication, that 
popular iUostrations are more frequently introduced, practical appli- 
eatioDs incessantly borne in mind, and such tables as seemed best 
odeakied to save tbe labour of Architects, Mechanics, and Civil 
EngineerB, inserted under their appropriate heads. Of these latter, 
several have been collected from former treatises, &c., but not a few 
have been either eompated or contributed expressly for this Common- 
place Book. 

In a work like this, it would be absurd to pretend to originality. 
Tbe plan, arrangement, and execution, are my own ; but the mate- 
rials have long been regarded, and rightly, as common property. It 
baa been my aim to reduce them into the smallest possible space, 
consiatratly with my general object ; but wherever I have found the 
work in this respect prepared to my hands, I have transcribed it into 
tbe following pages, with the usual references to the sources from 
whence it was taken. They who are conversant with the best writers 
on subjects of mixed nuitbematics and natural philosophy, will know 
that Smeaion^ Bobison^ Play/air^ Young^ Du Buat^ Leslie^ Hachette^ 
Blandy Tredgold^ &c., are authors who ought to be consulted, |n the 
preparation of a volume like this. I hope it will appear that I have 
duly, yet, at the same time, honourably, availed myself of the advan- 
tages which they supply. I have also made such selections from my 
own earlier publications as were obviously suitable to my present 
purpose ; but not so copiously, I trust, as to diminish the utility of 
those volumes, or to make me an unfair borrower even from myself. 

Besides our junior Civil Engineers, and the numerous Practical 
Mechanics who are anxious to store their minds with scientific facts 
and principles ; there are others to whom, I flatter myself, the fol- 
lowing pages will be found useful. Teachers of Mathematics, and 
those departments of natural philosophy which are introduced into 



TUl AUTHORS PRBFACB. 

our more respectable seminaries, may probably find tbis volume to 
occupy a convenient intermediate station between tbe merely popular 
exbibitions of tbe trutbs of Mecbanics, Hydrostatics, &c., and tbe 
larger treatises in wliicb tbe wliole cbain of inquiry and demonstra- 
tion is carefully presented link by link, and tbe successive portions 
firmly connected upon irrefragable principles. Wbile students wbo 
have recently terminated a scientific course, wbetber in our univer- 
sities, or otber institutions public or private, may, I would fain be- 
lieve, find in tbis Common -place Book an abridged repository of tbe 
most valuable principles and tbeorems, and of bints for tbeir applica- 
tions to practical purposes. 

Tbe only performances witb wbicb I am acquainted, tbat bear any 
direct analogy to tbis, are Martin's Young Students Memorial Bookj 
Jones's Synopsis Pcdmariorum Matkeseos^ and Brunton's Compendium 
of Mechanics; tbe latter of wbicb I had not seen until tbe present 
volume was nearly completed. Tbe first and last mentioned of tbese 
are neat and meritorious productions ; but restricted in tbeir utility 
by tbe narrow space into wbicb tbey are compressed. Tbe otber, 
written by the father of tbe late Sir William Jones, is a truly elegant 
introduction to tbe principles of Mathematics, considering tbe time 
in wbicb it was written (1706); but as it is altogether theoretical, 
and is, moreover, now becoming exceedingly scarce, it by no means 
supersedes tbe necessity^ for such I have been induced to regard it, of 
a Compendium like tbat wbicb I now offer to the public. 

In its execution I have aimed at no higher reputation than tbat of 
being perspicuous, correct, and useful ; and if I shall be so fortunate 
as to have succeeded in those points, I shall be perfectly satisfied. 

Olinthus Gregory. 

Boyal Military Academy, 
Woolwich, October 1st, 1825. 



In this new edition I have corrected a few errors wbicb had escaped 
my notice in the former impression. I have also made a few such 
additions and improvements as the lapse of time and tbe progress of 
discovery rendered desirable ; and such as will, I hope, give the work 
new claims on public approbation. 

July let, 1883. 



THE EDITOR'S PREFACE. 



In presenting to the public a new edition of Dr. Gregory's " Mathe- 
matics for Practical Men,** the Editor feels that a few introductory 
remarks are due from him to his readers. Not, indeed, to apologise 
for presenting to them again a work which they have already 
marked with their approbation, and which has from time to time 
been favourably noticed by other writers, but to offer some explana- 
tion of the numerous alterations which he has taken upon himself to 
make, and to state his reasons for having done so. 

In looking through the former edition, it was evident, that since 
the date of its publication, many of the subjects which it treated 
upon had been greatly improved and extended ; railways had been 
brought into successful operation ; the steam engine and machinery 
generally had undergone vast improvements ; and almost every branch 
of mechanical science had progressed in some degree ; it was there- 
fore found necessary, in order to render the present edition as exten- 
sively useful as its predecessors had been, that the work should 
undergo an entire revision. In doing this, it was found that so large 
a quantity of fresh matter would have to be given, as would swell 
the Tolume to an inconvenient size, and by a necessary increase of 
price, render the work less accessible to that important class — the 
working mechanic9*-for whose use it was so peculiarly adapted. To 
obviate, therefore, these disadvantages, a smaller type has been 
adopted, so that the work, although not much increased in its exter- 
nal dimensions, contains more than a third of its present bulk of new 
matter. 

In order to add to the usefulness of the work, and render it more 
generally convenient as a book of reference, a more orderly arrange- 
ment of its contents has been adopted, and several articles, which, 
although of much value, did not properly belong to the subject of the 
work, have been thrown into an Appendix ; in addition to which a 
Table of Contents has been given, showing at once the general ar- 



X KDIT0B8 PREPACB. 

rangement and division of the work, as also a very copious Index at 
the end. 

The Editor has endeavoured, in the following short review of the 
work in its present form, to point out the principal alterations or 
additions which have heen made : — 

The work has been divided into two distinct parts ; viz. 1st, Purb 
Mathematics, comprising chapters upon Arithmetic, Algebra, Geo- 
metry, Mensuration, Trigonometry, Conic Sections, and the Proper- 
ties of Curves; and, 2ndly, Mixed or Applied Mathematics, being 
the application of Mathematics to the general pursuits of the Engineer, 
the Mechanician, the Surveyor, &c., comprised in chapters upon 
Mechanics generally. Statics, Dynamics, Hydrostatics, Hydrody- 
namics, Pneumatics, Mechanical Agents, or Moving Powers, and the 
Strength of Materials. 

In the part upon Pure Mathematics : — 

In the First Chapter, upon Arithmetic, the article npon fractions 
has been extended, and rules given for the reduction of circulating 
decimals; and an entirely new section (IX.) is added upon Loga- 
rithmic Arithmetic, containing a description of the Logarithmic 
Tables given in the Appendix, with the method of using them, 
and roles for performing the ordinary processes of calculation by 
tlieir aid. 

The Second Chapter, on Algebra, has undergone a very careful re- 
vision. In Section V., on Involution, a table of roots and powers of 
monomials has been added ; and in the following section on Evolu- 
tion, a new rule for finding the roots of powers in general. In 
Section X. examples have been added of the method for determining 
the roots of equations. Section XII., on Fractional and Negative 
Exponents, is entirely new ; as is also nearly the whole of the next 
section on Logarithms ; to which have been added tables for converting 
Common Logarithms into Hyperbolic^ and vice versd. 

The Third Chapter, on Geometry, may almost be said to have been 
rewritten. The definitions and propositions have been arranged in a 
more orderly manner, and many additional ones have been added. 
No demonstration of the several propositions will be found, as the 
size of the work would not allow, nor did its practical character re- 
quire, that such should be given; but for the convenience of those 
who may wish to see them demonstrated, a reference has been made to 



BDIT0B8 PEBFACB. Xl 

Euclid, giving the number of the theorem and book in which the 
demonstration of the same proposition will be there found. 

The Fourth Chapter, on Mensuration, has also been almost entireiy 
rewritten : the tables of weights and measures have been put under a 
more conYenient form, and rules and formuIfiB giveft for the mensura- 
tion of all kinds of superficies and solids. 

In the Fifth Chapter, on Trigonometry, yery exteneiTe 2Vt^ofio- 
metrieal Formulm have \eea added, which cannot fail to be found of 
serrioe in the pursuits of Engineers and Surveyors. The second sec- 
tion is entirely new, being a description of the Tables of Logarithmic 
sines, cosines, tangents, cotangents, secants, and cosecants given in 
the Appendix, with the method of using them. 

The Sixth Chapter, on Conic Sections, has been almost entirely 
rewritten, and several additional formuls and problems relating to 
them have been added. 

The Seventh Chapter, on the Properties of Curves, has also been 
much extended ; the first section of definitions, and the latter part of 
the fourth section, on the Epicycloid, being entirely new. 

In the second division of the work, upon Mixsd Mathbm atigs:—- 

In the First Chapter, on Mechanics in General, many fresh defi- 
nitions have been added. 

In the Second Chapter, the whole of the definitions and formulfs 
in the first section, on Statical Equilibrium, are new. In the second 
section, the principle of parallel presmret is applied to determine 
the position of the center of gravity, and an example is given of the 
centro-baryc method of determining the superficies or solidity of 
surfaces or solids. In the third section, on the application of Statics 
to the equilibrium of structures, the formuln have been more conve- 
niently arranged, and tables have been added of the natural slope 
assumed by various kinds of materials, as also of the limiting angle 
and coefficient of friction of the various materials used in the con- 
struction of arches, &c, the latter extracted from Professor Moseley's 
work on the Mechanical Principles of Engineering. An article is 
also added upon Suspension Bridges, a subject of much importance 
to Engineers, from the general use into which they are being brought. 
Formultt are given for determining all their elements and propor- 
tions, the use of which is illustrated by an example. In the Third 
Chapter, on Dynamics, the definitions in the first section have been 
nuidi ezlended ; and at the commencement of the second section, on 



XU EDITORS PREFACE. 

the General Laws of Uniform and Variable Motion, it has been en- 
deayoured to put in a clear light the long disputed question relative 
to the momentum and vis viva of moving bodies. In this section, 
also, formulse have been added expressing the relation between the 
times, velocities, spaces, &c. of bodies in motion. In the third sec- 
tion some practical formulce are added for determining the proportions 
for the coniccd governor^ so generally used for regulating the velocity 
of steam engines and water wheels. In the fifth section, on the 
Mechanical Powers, their true nature is explained, in order to pre- 
yent any misconception which might be occasioned by their designa- 
tion ; and a table is added exhibiting the ratio in each, between the 
power exerted and the effect produced. 

In the Fourth Chapter, on Hydrostatics, the fifth section, on Capil- 
lary Attraction, is entirely new. 

In the Fifth Chapter, on Hydrodynamics, several additional for- 
malse are given, in the second section, for determining the velocity 
and discharge through conduit pipes ; and the whole of the formulse 
have been arranged in a more orderly manner, and numbered for the 
convenience of reference. 

In the Sixth Chapter, on Pneumatics, at the conclusion of the 
first section, a very exact formul«B of Mr. Galbraith's is given, for 
determining the velocity of sound, and for measuring distances by the 
same. 

In the Seventh Chapter, on the Mechanical Agents, the third sec- 
tion, on the Steam Engine, is entirely new. The nature and pro- 
perties of steam are explained, and formulse and rules are given, and 
illustrated by examples, for determining its pressure or temperature. 
The subject of the specific heat of steam is also explained, and for- 
mulse given for determiHing it from the pressure; and a table is 
added of the pressure, temperature, and specific volume of steam, 
from 5 lbs. to 1 50 lbs. upon the square inch. A general description 
of the mode of action of the steam engine is given, which is fol- 
lowed by an investigation of its general theory upon the principles 
first laid down by the Comte de Pambour. The various kinds of 
engines employed are then classified, and described separately ; the 
principles previously educed are applied to each ; and formulse are 
deduced for the velocity of the engine, the quantity of steam used, 
and the useful effect of the engine, for each particular case, which 
are given both in a general and a more practical form. In this por- 
tion of the work, in treating of engines acting expansively, it was 



EDITORS PREFACE. Xli) 

found imposnble to dispense with the use of the Differential Calculus, 
the introduction of which had heen studiously avoided, in order that 
the work might be accesable to those who had not entered upon the 
higher branches of Mathematics. The use of the Differential Cal- 
culus, howeyer, in the present case, while it was necessary for the 
proper investigation of the subject, will not, it is hoped, in any way 
abridge the usefulness of the work to those who are not conversant 
wiUi its principles, since the results obtained are expressed in the 
ordinary form, while for those who do understand them, it was 
thought that it would prove much more satisfJEictory to them to be 
enabled to examine and satisfy themselves of the truthfulness of the 
several investigations. 

The Eighth Chapter, on the Strength of Materials, has been 
almost entirely rewritten. This portion of the work has been 
rendered much more valuable by the introduction of the results of 
Professor Hodgkinson and Mr. Cubitt's researches upon this subject, 
«o important to the Civil Engineer and Architect. 

The woodcuts have been replaced by copperplate engravings, and 
several additional plates have been given, for three of which the Editor 
is indebted to Mr. R. A. Rumble. 

An Appendix has been added, containing a Table of Logarithmic 
Differences, in a new, and, it is hoped, more convenient form than 
that usually adopted, and Tables of the Logarithms of Numbers, and 
of Logarithmic sines, tangents, &c. : these tables will be found of 
great utility to Engineers and others, much engaged in calculations. 
It also contains a new Table ( VL) of various useful numbers, with 
their logarithms, and a Table (XI.) of the weight of materials fre- 
ijuently employed in construction. The remaining portion of the 
Appendix consists of matter which stood in the body of the former 
edition of the work, but which has now been put into the Appendix, 
because, although too valuable to be omitted, it did not properly be- 
long to, but was only incidentally connected with, the subjects 
therein treated. 

The Editor has thought it t>nly doing justice to the late Dr. Ore- 
gory to point out thus in detail the extent of the alterations which he 
has made, in order that the public may know how far each is re- 
sponsible for the work in its present form, and to prevent either 
blame being attached to Dr. Gregory, or credit given to the Editor, 

where both might have been misplaced. 

H. L. 
London, 
21ft October, 1847. 



CONTENTS. 



PART I.— PUKE MATHEMATICS. 

CHAPTER I. 
Arithmetic. 

PACK 

Sect. 1. Definitions and Notation. 1 

2. Addition of Whole Numben 5 

S. Subtraction of Whole Numben 6 

4. Multiplication of Whole Numben 7 

5. DiTinon of Whole Numben 10 

Proof of the fint Four Rules of Arithmetic . . . .Id 

6. Vulgar Fractions 18 

Reduction of Vulgar Fractions . . . . . .14 

Addition and Subtraction of Vulgar FVactions . .17 

Multiplication and Division of Vulgar Fractions . . 17 

7. Decimal Fractions 18 

Reduction of Decimals 19 

Addition and Subtraction of Decimals 22 

Multiplication and Division of Decimals 22 

8. Complex Fractions used in the Arts and Commerce . .23 

Reduction 28 

Addition 24 

Subtraction and Multiplication 25 

Division * . .26 

Duodecimals 27 

9. Powen and Roots 28 

Evolution 29 

10. Ph)portion 82 

Rule of Three 84 

Determination of Ratios 87 

11. Logarithmic Arithmetic ^ 

Use of die Tablet 42 

Multiplication and Division bv Logarithms . . • . ^ 
Proportion, or the Rule of l^ree, by Logarithms . . 46 

Evolution and Involution by Lo^urithms .47 

12. Properties of Numben 49 

CHAPTER IL 

Algbbba. 

8acr. I. Definitions and NoCatioti A2 

2. AdfidoD and Subtraction ^ 



XVI CONTENTS. 

PAGE 

Sect. S. Multiplication 56 

4. Division 58 

5. Involution 60 

6. Evolution 63 

7. Surds 65 

Reduction 65 

Addition, Subtraction, and Multiplication .... 68 

Division, Involution, and Evolution 69 

8. Simple Equations 70 

Extermination 73 

Solution of Genera] Problems 75 

9. Quadratic Equations 77 

10. Equations in General 81 

11. Progression 84 

Arithmetical Progression 85 

Geometrical Progression 86 

12. Fractional and Negative Exponents 88 

13. Logarithms 90 

14. Computation of Formulae 95 



CHAPTER in. 
Geometry. 

Sect. 1. Definitions 98 

2. Of Angles, and Right Lines^ and their Rectangles . . .99 

a Of Triangles 100 

4. Of Quadrilaterals and Polygons 102 

5 Of the Circle, and Inscribed and Circumscribed Figures . .104 

6. Of Planes and Solids 109 

7. Practical Geometry 112 



CHAPTER IV. 
Men&uration. 

Sect. 1. Weights and Measures 119 

1. Measures of Length 120 

2. Measures of Sur&e 12] 

3. Measures of Solidity and Capacity 121 

4. Measures of Weight 123 

5. Angular Measure 124 

6. Measure of Time 124 

Comparison of English and French Weights and Measures . 125 

2. Mensuration of Supeirficies ] 27 

3. Mensuration of Solids 130 



CHAPTER V, 

Trigonometry. 

Sect. 1. Definitions and Triffonometrical Formulae 134 

2. Trigonometrical Tables 139 



CONTENTS. XVU 

PAGE 

SxcT. Sw Geoenl Propoeitkms 146 

4. Solution of the Casei of Plane Triangles 148 

Rifffat-angled Plane Triangles 151 

5. On Uie application of Trigonometry to Measuring Heights and 

Distances 152 

Detenninatipn of Heights and Distances by Approximate Me- 

chanical Methods 156 



CHAPTER VL 
CoNie Sections. 

8xcT. 1. Definitions 162 

2. Properties of the Ellipse 165 

Problems relating to the Ellipse 167 

3. Properties of the Hyperiwla 168 

Problems relatine to Hyperbolas 170 

4. Properties of the Parabola 171 

Problems relating to the Parabola 173 

CHAPTER VIL 

PROFEBTISS OF CUETES. 

Sect. 1. Definitions 175 

2. The Conchoid 176 

a The Cissoid 177 

4. The Cycloid, and Epicycloid 178 

6. The Quadratriz 179 

& The Catenaiy 180 



tenaiy . 
I of Relat 



Tables of RelatioDs of Catenarian Curves .... 185 



PAET n.— MIXED MATHEMATICS. 

CHAPTER I. 
Mechanics in Oxneeal 187 

CHAPTER II. 

Statu». 

Sect. 1. Statical Eouilibrium 180 

2. Center of Gravi^ 102 

3l General application of the Prmciples of Statics to the Equili- 
brium 01 Structures 196 

Equilibrium of Piers or Abutments 196 

Pressure of earth against Walls 198 

Thickness of Walls 201 

Bquilibrinm of Polygons 202 

Stability of Arches .' . s . .205 

Bqpdilicium of Suspension Bridges 207 

h 



XVUl CONTENTS. 

CHAPTER III. 
Dynamics. 

PAGE 

Sect. 1. Oenenl Definitions 211 

2. On the General Laws of Uniform and Variable Motion 212 

Motion uniformly Accelerated 214 

Motion of Bodies under the Action of Gravity . .214 

Motion over a fixed Pulley 216 

Motion on Inclined Planes 217 

3. Motions about a fixed Center, or Axis 220 

Centers of Oscillation and Percussion 220 

Simple and Compound Pendulums 221 

Center of Gyration, and the Principles of Rotation . 229 

Central Foroes 232 

Inquiries connected with Rotation and Central Forces . 234 

4. Percussion or Collision of Bodies in Motion .... 236 

5. On the Mechanical Powers 239 

Levers 240 

Wheel and Axle 242 

Pulley 243 

Inclined Plane 244 

Wedge and Screw 243 



CHAPTER IV. 

Hydrostatics. 

Sect. 1. General Definitions 248 

2. Pressure and Equilibrium of Non-elastic Fluids .... 249 

3. Floating Bodies 254 

4. Specific Gravities 256 

5. On Capillary Attraction 258 



CHAPTER V. 

Hydrodynamics. 

Sect. 1. Motion and Effluence of Liquids 260 

2. Motion of Water in Conduit Pipes and Open Canals, over 

Weirs, «6C 262 

Velocities of Rivers 265 

3. Contrivances to Measure the Velocity of Running Waters . . 273 



CHAPTER VI. 

Pneumatics. 

Sect. 1. Weight and Equilibrium of .Air and Elastic Fluids . . . 278 

2. Machines fur Raising Water by the Pressure of the Atmosph ere 281 

3. Force of the Wind 292 



CONTENtS. Xix 

CHAPTER VIL 
BIechanical Agents. 

PAGE 

Sect. 1. Water as a Mechanical Af^nt 294 

2. Air as a Mechanical Agent 297 

Coulomb's Experiments 801 

dw Mechanical Agents depending upon Heat The Steam Engine . 802 
Table of Pressure and Temperature of Steam . . .811 

General Description of the Mode of Action of the Steam Engine 812 

Theory of the Steam Engine 816 

Description of the various kinds of Engines, and the Formulffi 

for calculating their Power 822 

Practical application of the foregoing Forroulie . 848 

4. Animal Strength as a Mechanical Agent 848 



CHAPTER VIIL 
Strength of Materials. 

SaT. 1. Results of Experiments, and Principles upon which they should 

be practically applied 870 

2. Strength of Materials to Resist Tensile and Crushing Strains 874 

Strength of Columns 879 

3. Elasticity and Elongation of Bodies subjected to a Crushing or 

Tensile Strain 881 

4. On the Strength of Materials subjected to a Transverse Strain 885 

Longitudinal form of Beam of uniform Strength . 886 

Transverse Strength of other Materials than Cast Iron . 887 
The Strength of Beams according to the manner in which the 

Load is distributed 888 

a. Elasticity of Bodies subjected to a Transverse Strain . . 888 

6. Strength of Materials to resist Torsion 892 



APPENDIX. 

L Table of Logarithmic differences 1 

n. Table of Logarithms of Numbers, from 1 to 100 .... 1 

HL Table of Logarithms of Numbers, from 100 to 10,000 ... 1 
IV. Table of Logarithmic Sines, Tangents, Secants, &c. .19 

V. Table of Usoiil Factors, extending to several places of Decimals . 64 

VL Table of various Useftil Numbers, with their Logarithms ... 66 
VIL A Table of the Diameters, Areas, and Circumferences of Circles 

and also the Sides of Equal Squares 66 

VIII. Table of the Relations of the Arc, Abscissa, Ordinate and Subnormal, 

in the Catenary 78 

IX. Tables of the Lengths and Vibrations of Pendulums ... 76 

X. Table of Specific Gravities 80 

XL Table of Weight of Materials frequently employed in Construction . 82 

XIL Principles of Chronometers 82 

XIII. Select Mechanical Expedients 89 

XIV. Obaervarions on the Effect of Old London Bridge on the Tides, &c. 92 
XV. Professor Parish on Isometrical Perspective -Wi 



LIST OF PLATES. 



Platb L' 
IL 

IIL V General Diafframs. 
IV. ^ 
V. 

VI. Detiili of a Breast Water-wheel. 
VI L Fenton, Murray, and Wood*a Steam Engine. 
VII L Higb-prenure Expansive Engine, by Middleton. 
IX. Details of Steam Engines. 
X. Longitudinal Section of a Locomotive Engine. 
XL Two Transferee Sections of the Same. 
XIL Details of various Engines. 
XIIL Isometrical Perspective. 



COMMON-PLACE BOOK, 



ETC., ETC. 



PART I. 

PURE MATHEMATICS. 



CHAP. I. 

ARITHMETIC. 

Sect. I. Definitions and Notation. 

Arithmetic is the science of numbers. 

We give the name of number to any assemblage of units^ or of any 
parts of an assumed unit; a unit being the quantity which, among 
all those of the same kind, forms a whole which may be regarded as 
the hose or element. ' Thus, when we speak of one kouse^ one guinea^ ^ 
we speak of units^ of which the first is the thing called a house, the 
fiecond that called a guinea. But when we w,y four houses^ ten 
gmnea^^ three quarterM ofa-guinea^ we speak of numbers^ of which 
the first is the unit house repeated four times; the second is the unit 
QHinea repeated ten times; the third is the fourth part of the unit 
guinea ^peated three times. 

In cv«y particular classification of numbers, the unit is a measure 
taken aroitrarily, or established by usage and convention. 

Numbers formed by the repetition of an unbroken unit are called 

vhole numbers^ or integers^ as seven miles, thirty shillings: those 

which are formed by the assemblage of any parts of a unit are called 

fractional numbers^ or %\mi^\yf radians; as two thirds of a yard, three 

eighths of a mile. 

When the unit is restricted to a certain thing in particular, as one 
man^ one horse^ one pounds the collection of many of those units is 
called a concrete number, aa ten men^ twenty horses^ fifty pounds. 
But if the unit does not denote any particular thing, and is expressed 
^mply by one^ numbers which are constituted of such units are deno- 
minated dieerete or ahstracty as five^ ten, thirty. Hence, it is eyident 
that abstract nmnbers can only be compared with their unit, as con- 

B 



ARITHMETIC : NOTATION. 



[part I. 



Crete numbers are compared with, or measured by, theirs; but that it 
is not possible to compare an abstract with a concrete number, or a 
concrete number of one kind with a concrete number of another ; for 
there can exist no measurable relations but between quantities of the 
same kind. 

The series of numbers is indefinite; but only the first nine of them 
are expressed by different characters, called figures; which are as 
follows : — 

Names, one, two, three, four, ^sc^ six, seven, eight, nine. 

Figures. 1, 2, 3, 4, 5, C, 7, 8, 9. 

These are called sig?iificant figures, in contradistinction to another 
character employed, namely 0, called the cypher or zero; which has 
no particular value of itself, but by its positmi is made to change the 
value of any significant figures with which it is connected. 

In the system of numeration now generally adopted, and which is 
borrowed from the Indians*, an infinitude of words and characters is 

* As the Roman notation is not unfrequently met with, especially in dates, 
we subjoin the following brief account of it: — The Romans employed only 
seven numerals, being the following capital letters of their alphabet ;— viz. I, 
for one ; V, for five; X, for ten; L, for fifty ; C, for an hundred ; D, for five 
hundred; M, for k thousand ; and for expressing any intermediate or greater 
numbers they employed various repetitions and combinations, the principles of 
which are shown in the following examples : — 

As often as any character is repeated, so 
many times is its value repeated. 

A less character before a greater diminishes 
\ its value. 

( A less character after a greater increases 
( its value. 



I 

ir 
III 



4 = nil or IV 



5 

6 < 

7 •- 
8 
9 

10 
40 
50 

m 

100 
500 

1000 
2000 
5000 

0000 

10,000 

50,000 

00,000 

100,000 

1,000,000 

2,000,000 

&c. 



V 

VI 

VII 

VIII 

IX 

X 

XL 

L 

LX 

C 

Dor lo 

M or Clo 

MM 

V or loD 

yi 

X or CCI33 
W I J33 
LX 

CorCCCl033 
M or CCCCI3303 
MM 
&c. 



5 For every f) annexed, this becomes ten 
( times as many. 

\ For every C and ^), placed one at each 
( end, it becomes ten times as much. 

S A bar over any num1)er increases its value 
\ 1000-fold. 



CHIP. I.] arithmetic: notation. 3 

avoided, by a simple yet most ingenious expedient, which is this : — 
(urjifyHre placed to t/te left of another assumes ten times the wUue 
(to \i rtoM have if it occupied the place of the latter. 

Thus, to express the number that is the sum of 9 and 1, or ten 
units, (called ten^) we place a 1 to the left of a 0, thus 10. So again 
the sum of 10 and 1, or eleven^ is represented by 11 ; the sum of 11 
ud 1, or of 10 and 2, (called twelWy) is represented by 12 ; and so 
on for tiwieen, fourteen^ fifteen^ &c., denoted respectively by 13, 14, 
15, &c, the figure 1 being all along equivalent to ten^ because it 
oecapies the second rank. 

la like manner, twenty^ twenty^one^ twenty-two^ &c., are repre- 
sented by 20, 21, 22, because the 2 in the second rank is equivsJent 
to twice ten, or tteenty. And thus we may proceed with respect to 
the numbers that fall between twenty and three tens or ^irty (30), 
four tens or forty (40), five tens or Jifily (50), six tens or sixty (60), 
RTen tens or seventy (70), eight tens or eiyhty (80), nine tens or 
iit)ie^(90). After 9 are added to the 90 (ninety), numbers can no 
longer be expressed by two figures, but require a third rank on the 
left hand of tbe second. 

Tbe figure that occupies the third rank, or of hundredths^ is ex- 
pressed by the word hundred. Thus 369, is read three hundred and 
sixtj-nine ; 428, Is read four hundred and twenty-eight ; 837, eight 
kundred and thirty-seven : and so on for all numbers that can be re- 
presented by three figures. 

Bat if the number be so large that more than three figures are 
required to express it, then it is customary to divide it into periods of 
tkree figures each, reckoning from the right hand towards the \eh, and 
to distinguish each by a peculiar name. The second period is called 
tbat of thousands^ the third that of millions^ the fourth that oi milliards 
or billions*^ the fifth that of trillions^ and so on ; the terms units, tens, 
ud hundreds, being successively applied to the first, second, and 
third ranks of figures from the right towards the left, in each of these 
periods. 

Thus, 1,111, is read one thousand one hundred and eleven. 

23,456, twenty-three thousands, four hundred and fifty-six. 

421,835, four hundred and twenty-one thousands, eight hundred 
•nd thirty-five. 

732,846,915, seven hundred and thirty-two millions, eight hundred 
tnd forty-six thousands, nine hundred and fifteen. 

The manner of estimating and expressing numbers which we have 
here described is conformable to what is denominated the decimal 
notation. But, besides this, there are other kinds invented by philo- 
sophers, and others indeed in common use : as the duodecimal^ in 

' It has been cascmnary in England to give the name of LiHums to millions 
of millions, of triUknu to millions of millions of millions, and so on : but the 
method here given of dividing numbers into periods of three figures instead of 
six, it universal (m the Continent ; and, as it seems more simple and uniform 
tian the other, I have adopted it. 

B 2 



4 arithmetic: notation. [part i. 

which every superior name contains (tcelve units of its next inferior 
name ; and the sexagesimal^ in which sixty of an inferior name are 
equivalent to one of its next superior. The former of these is em- 

{)loyed in the measurement and computation of artificers' work ; the 
atter in the division of the circle, and of an hour in time. 

To the head of notation we may also refer the explanation of the 
principal symbols or characters employed to express operations or 
results in computation. Thus, 

The sign + (plus) belongs to addition, and indicates that the 
numbers between which it is placed are to be added together. Tlius, 

5 + 7 expresses the sum of 5 and 7, or that 5 and 7 are to be added 
together. 

The sign — (mintis) indicates that the number which is placed after 
it is to be subtracted from that which precedes it. So, 9—3 denotes 
that 3 is to be taken from 9. 

The sign ^^ denotes diffei-ence^ and is placed between two quan- 
tities when it is not immediately evident which of them is the greater. 

The sign x {into\ for multiplication, indicates the product of two 
numbers between which it is placed. Thus 8x5 denotes 8 times 5, 
or 40. 

The sign -f- {hif)^ for division, indicates that the number which 
precedes it is to be divided by that which follows it ; and the quotient 
that results from this operation is often represented by placuig the 
first number over the second with a small bar between them. Thus, 
15 -r- 8 denotes that 15 is to be divided by 8, and the quotient is ex- 
pressed thus y . 

The sign =, two equal and parallel lines placed horizontally, is 
that of equality. Thus, 2 -|- 3 + 4 = 9, means that the sum of 2, 3, 
and 4, is equal to 9. 

Inequality is represented by two lines so drawn as to form an angle, 
and placed between two numbers, so that the angular point turns to- 
wards the least. Thus, 7^4, and A > B, indicate that 7 1^ greater 
than 4, and the quantity represented by A greater than the quantity 
represented by B ; and, on the other hand, 3 < 5 and C < D indicate 
that 3 is less than 5, and C less than D. 

Colons and double colons are placed between quantities to denote 
their proportionality. So, 3 : 5 : : 9 : 15, signifies that 3 are to 5 
as 9 are to 15, or | zi yj. 

The extraction of roots is indicated by the sign v/, with a figure 
occasionally placed over it to express the degree of the root ; or by a 
fraction (having unity for its numerator, and the figure expressing the 
degree of the root for its denominator) placed above and to the right 
of the quantity to have its root extracted ; thus, v/ 4 or 4*, signifies 
the square root of 4 ; V27, or 27*, the cube root of 27; V 16 or 16*, 
the fourth or biquadrate root of 1 6. 

The raising of powers is expressed by a whole number similarly 
placed, the figure denoting the, power to which the quantity is to be 
raised ; thus 6'. signifies the square of 6 ; 8', the cube of 8 ,* and 3\ 



CHAP. I.] ADDITION OF WHOLE NUMBERS. 5 

the fourth power of 3. The figures thus used to indicate the power, 
whether whole or/ractional numbers, are termed indices or exponents. 

When both operations are to be successively performed upon a 
quantity, that is, when some root is to be extracted, and then that root 
to be raised to some different power, the operation is very simply ex- 
pressed by a fraction placed as before, the denominator of which indi - 
cates the root to be extracted, and the numerator the power to which 
that root is to be afterwards raised. Thus 27*, denotes that the cube 
root of 27 is to be extracted, and that the root so obtained is to be 
squared, or raised to the second power. It is immaterial which 
operation is first performed, for the result would be the same whether 
we first extracted the cube root, as above, and then squared it^ or 
whether we squared the number first, and then extracted the cube root 
of the power so obtained. 

Although the above signs * are principally employed in Algebra 
and the higher branches of Mathematics, they are given here, as their 
use in Arithmetic frequently affords brevity without a sacrifice of 
perspicuity. 



Sect. II. Addition of Whcle Numbers, 

Addition is the rule by which two or more numbers are collected 
into one aggregate or sum. 

Suppose it were required to find the sum of the numbcre 3731, 
349, 12487, and 54. It is evident that if we computed separately . 
the sums of the units, of the tens, of the hundreds, of the thousands, 
&c^ their combined results would still amount to the same. We 
should thus have 15 thousands + 14 hundreds + 20 tens + 21 units, 
or 15000 + 1400 + 200 + 21 ; operating again upon these, in 
like manner, rank by rank, we should have 1 thousands + 6 thou- 
sands + 6 hundreds + 2 tens + 1, or 16621, which is the sum 
required. 

But the calculation is more commodiously effected in the following 
manner : — 

Rtde. — Place the given numbers under each other, so that units 
stand under units, tens under tens, hundreds under hundreds, &c. 
Add up all the figures in the column of units, and observe for every 
ten in its amount to carry one to the place of tens in the second column, 
putting the overplus figure in the first column. Proceed in the same 
manner with the second column, then with the third, and so on till 
alJ the columns be added up : the figures thus obtained in the 
several amounts indicate, according to the rules of notation, the sum 
required. 

Nate. — Whether the addition be conducted upwards or downwards, 
the result will be the same ; but the operation is most frequently con- 
ducted by adding upwards. 

* There are other signs employed in the processes of Algebra, an explanation 
of which will Ue found at page 52, st icq. 



6 SUBTBACTION OF WHOLE NUMBBBS. [PABT I. 

Example. — Taking the same numbers as before, and 3731 

disposing them as the rule directs, we have 4 + 7 + 9 + 349 

1 =z 21, of which we put down the 1 in the place of uniU^ 12487 

and carry the 2 to the tens : then 2-h5 + 8 + 4-f3= 54 

22, of which we put down the left hand 2 in the place of 

tens^ and carry the other to the hundreds : then 2 + 4 + 1662 1 

3 + 7 = 16, of which the 6 is put in the hundreds^ and =*^- 

the 1 carried to the thousands. This progress continued will give the 
same sum as before. 

Other Examples, 



57 

762 

5389 

97615 


III. 


6475 
9830 
2764 
5937 


77756 
3388 
9763 

90257 


10376786 

789632 

1589 

73 


103823 


25006 


181164 


11168080 


Sect. 


Subtraction Oj 


f mole Numbers. 



Subtraction is the rule by which one number is taken from 
another, so as to show the difference, or excess. 

The number to be subtracted or taken away is called the subtrahend ; 
the number from which it is to be taken, the minuend; and the quan- 
tity resulting, the remainder. 

Rule. — Write down the minuend, and beneath it the subtrahend, 
units under units, tens under tens, and so on. Then beginning at the 
place of units, take each figure in the subtrahend from its correspond- 
ing figure in the minuend, and write the difference under those figures 
in the same rank or place. 

But if the figure in the subtrahend be greater than its corresponding 
figure in the minuend, add ten to the latter, and then take the figure 
in the subtrahend from the sum, putting down tlie remainder, as be- 
fore ; and in this case add 1 to the next figure to the le/t in the sub- 
trahend, to compensate for the ten borrowed in the preceding place. 
Thus proceed till all the figures are subtracted. 

Example: . . Minuend 26565874 
Subtrahend 9853642 



Remainder 16712232 



Here the five figures on the right of the subtrahend are each less 
than the corresponding figures in the minuend, and may therefore be 
taken from them, one by one. But the sixth figure, viz. 8, cannot be 
taken from the 5 above it. Yet, as a unit in the seventh place is 
equivalent to 10 in the sixfhy this unit borrowed (for such is the 



CHAP, I.] MULTIPLICATION OP WHOLB NU1IBBB8. 7 

technical word here employed) makes the 5 become 15. Then 8 
taken from 15 leaves 7, which is put down ; and 1 is added to the 9 
in the seventh place of the subtrahend, to compensate or balance the 
1 which was borrowed from the seventh pkwjc in the minuend. Re- 
course must be had to a like process whenever a figure in the sub- 
trahend exceeds the corresponding one in the minuend. 

Other Examples, 

From 8217 From 44444 Take 21498 Take 45624 

Take 3456 Take 3456 From 76262 From 80200 



Remains 4761 Remains 40988 Remains 54764 Remains 34576 

iVbte. — Although it is customary to place the minuend above the 
subtrahend, this is not absolutely necessary. Indeed, it is often con- 
venient in computation to find the difference between a number and 
a greater that naturally stands beneath it : it is, therefore, expedient 
to practise the operation in both ways, so that it may, liowever it 
occurs, be performed without hesitation. 



Sbct. IV. MuUiplication of Whole Numbers. 

Multiplication of whole numbers is a rule by which we find 
what a given number will amount to when it is repeated as many 
times as are represented by another number *. 

The number to be multiphed, or repeated, is called the mtUiipli- 
candj and may be either an abstract or a concrete number. 

The number to be multiplied by is called the multiplier^ and mtist 
be an abstract number, because it simply denotes the number of times 
the multiplicand is to be repeated. 

Both multiplicand and multiplier are called yac^or«. 

The number that results from the multiplication is called the pro- 
duct. 

Before any operation can be performed in multiplication, the 
learner must commit to memory the following table of products, from 
2 times 2, to 12 times 12. 

* This definition, though not the most scientific that might be given, is placed 
Ymbt^ because others depend, implicitly if not explicitly, on proportion, and 
Iharafore cannot logically be introduced thus early in the course. 



MULTIPLICATION OF WHOLE NUMBERS. 



[part I. 



times 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 
24 


2 


4 


6 


8 


10 


12 


14 


16 


18 


20 


22 


3 
4 


6 


9 


12 


15 


18 


21 


24 


27 


30 


33 


36 


8 


12 


16 


20 1 24 


28 


32 


36 


40 


44 


48 


5 


10 


15 


20 


25 


30 


35 


40 


45 


50 


55 


60 


6 


12 


18 


24 


30 


36 


42 


48 


54 


60 


66 


72 


7 


14 


21 


28 


35 


42 


49 


56 


63 


70 


77 
88 


84 
96 


8 


16 


24 


32 


40 


48 


56 


64 


72 


80 


9 


18 


27 


36 
40 


45 


54 


63 


72 


81 


90 


99 


108 


10 


20 


30 


50 


60 


70 


80 


90 


100 


110 


120 


11 


22 


33 


44 


55 


66 


77 


88 


99 


110 


121 


132 
144 


12 


24 


3G 


48 


60 


72 


84 


96 


108 


120 


132 



It is very advantageous, in practice, to have this table carried on, 
at least intellectually, to 20 times 20. All the products to this extent 
are easily remembered. 

The learner will perceive that in this table 7 times 5 is equal to 
5 times 7, or 7 X 5 =: 35 = 5 X 7. In like manner that 
8 X 3 = 24 = 3 X 8, 4 X 11 = 44 z= 11 X 4, and so of other 
products. This is often made a subject of formal proof, as well as that 
3x5x8 — 3x8x5 = 5x3x8 = 5x8x3, &c. But 
to attempt the demonstration of things so nearly axiomatical as these 
is quite unnecessary. 

Previously to exhibiting the rules for performing multiplication, let 
us take a simple example, and multiply 4827 by 8. Here 
placing the numbers as in the margin, and multiplying in 
their order 7 imits by 8, 2 tens by 8, 8 hundreds by 8, 4 
thousands by 8, the several products are 56 units, 16 tens, 
6.4 hundreds, 32 thousands : these placed in their several 
ranks, according to the rules of notation, and then added up, 
give for the sum of the whole, or for the product of 4827 
multiplied by 8, the number 38616. 



4827 
8 

56 
16 
64 
32 

38616 



CHAP. I.] MULTIPLICATION OP WHOLE NUMBERS. 9 

Or the same example may be worked thus : — 
8 X 7 = 56 \ 

Sx 800= eJoo/ ^^^"^^^«^I^^^°^^3^i*^^ 

8 X 4000 = 32000 i f™® '" ^"^^^ ^ ^''' 
I fore. 

38G16 /' 

Cask I. — To multipltf a number^ consisting of several figures^ by a 
Mumher not exceeding 1 2. 

Ruk. — Multiply each figure of the multiplicand by the multiplier, 
beginning at tlie units ; write under each figure the units of the pro- 
duct, and carry on the tens to be added as units to the prodqct fol- 
lowing. 

Examples, 



Mnltiplv 


4827 


218043 


440052 


8765400 


Bv ' 


8 


9 


11 


12 



Products 38616 1962387 4840572 105184800 



Case II. — To perform multiplication token each /actor exceeds 1 2. 

Rule. — Place the factors under each other (usually the smallest at 
bottom), and so that units stand under units, tens under tens, and so 
on. Multiply the multiplicand by the figure which stands in the unit's 
place of the multiplier, and dispose the product so that its unit's place 
^hftll stand under the unit of the multiplicand ; then multiply succes- 
sixely by the figure in the place of tens, hundreds, &c., of the multi- 
plier, and place the first figure of each product under that figure of the 
moltiplicr which gave the said product. The sum of these products 
will be the product required. 

Example, 

Multiply 8214356 by 132. 

Multiplicand 8214356 
Multiplier 132 

8214356 X 2 = 16428712 

8214356 X 3 tens = 24643068 

8214356 X 1 hundred = 8214356 

8214356 X 132 = 1084294992 



10 DIVISION OF WHOLB NUMBBB8. [PART I. 

Other Example$. 
Multiply 821436 Multiply 8210075 

by 672576 by 420306 



4928616 49260450 

5750052 24630225 

4107180 16420150 

1642872 32840300 

5750052 



4928616 Product 3450743782950 



Product 552478139136 



Note, — Multiplication may frequently be shortened by separating 
the multiplier into its component parts or factors, and multiplying by 
tbem in succession. Thus, since 132 times any number are equal to 
12 times 11 times that number, the first example may be performed 
in this manner : 

Multiply 8214356 



by 11 



Here one line of multi- 



And this product 90357916 ,» plication, and one of 
by 12 i addition, are saved. 

Product as before 1084294992 / 



So, again, the multiplier of the second example, viz. 672576, 
divides into three numbers, 600000, 72000, and 576; where, omitting 
the cypher, we have 72 =: 12 X 6, and 576 = 8 X 72. Hence 
the operation may be performed thus: — 

Multiplicand 821436 

Multiply by 6 in the 6th place. 

4928616 
Previous product X 12 ... 59143392 for 72 thousands. 
Second product X 8 473147136 for 576 units. 

Same product as before 552478139136 : three lines saved. 

Other modes of contraction will appear as we proceed. 

Sbct. V. Division of Whole Numbers. 

Division is a rule by which we determine how often one number 
is contained in another. Or, it is a rule by which, when we know a 
product and one of the factors which produced it, we can find the 
other. 

The number to be divided in called the dividend ; that by which it 



CHAP. I.] ]>I VISION OP WHOLB NUMBBRB. 11 

is diTided, the divisor; and that which results from the divisioD, the 
([ua^tnt. When division and multiplication are regarded as reciprocal 
operations, the dividend is equivalent to the product^ the divisor is 
equiTalent to the tntdtiplier^ and the quotient is equivalent to the mtd- 
tipUeand. 

Rule. — Draw a curved line both on the right and left of the dividend, 
and place the divisor on the left ; then find the number of times the 
dirisor is contained in as many of the left-hand figures of the dividend 
as are jost necessary, and place that number on the right. Multiply 
the divisor by that number, and place the product under the above- 
mentioned figures of the dividend. Subtract the said product from 
that part of the dividend under which it stands, and bring down the 
Bext figure of the dividend to the right of the remainder. Divide the 
remainder thus increased, as before ; and if at any time it be found 
less than the divisor, put a cypher in the quotient, bring down the 
next figure of the dividend, and continue the process till the whole is 
finished : the figures thus arranged will be the quotient required. 

Examples, 

Divide 743256 by 324. 
Dividend. 
Divisor 324)743256(2294 Quotient. 

648 Divisor 324 
Quotient 2294 



648 

3045 
2916 


Pro 

131)135076(1031^^^: ' 
131 


1296 
2916 
648 
648 


1296 
1296 

Remain 

71)29754(419^ 
284 


►of 743256 


In these two ex% 
amples the num- 
bers which re- 
main are placed 
ot^ their respect- 
ive divisors, and 
"attached to the 
quotients ; the 
meaning of which 
will be expbiined 
when we treat of 


135 
71 

644 
639 

5 Remain. 


407 
393 

146 

131 



15 Remain. 



Note.-^Wheu the divisor does not exceed 12, the operation may 
readily be perfoimed in a single line; as will appear very evident if 



12 



DIVISION OF WHOLE NUMBERS. 



[PABT I. 



tbc following example be compared with the two methods of working 
the first example in multiplication. 

Divide 38616 by 8. 
8)38616(4827 Dividend 38616 

32 Divisor 8 



66 
64 



Quotient 4827 



21 
16 

56 
56 



Here 38 contains 8 four times, leaving a remainder 
of 6; these carried as 6 tens to the next 6, make 
66, which contains 8 eight times, leaving 2, which 
carried as 2 tens to the next figure 1, make 21 : and 
so of the rest. 



In division, also, upon the same 
principle as in multiplication, the la- 
bour may often be abridged by taking 
component parts of the divisor. Thus, 
in the first example, the divisor is equal 
to 4 times 81, or 4 times 9 times 9. 
Hence the dividend may be divided by 
4, 9, and 9, successively, as in the 
margin, and the result will be the same 
as before. 



Divide 743256 
by 4 



this quotient 185814 
by 9 



and this 20646 
by 9 



Quotient 2294 



Since 25 is a fourth part of 100, and 125 the 8th part of 1000, 
it will be easy to multiply or to divide by either of these numbers iu 
a single line — thus. 



To multiply 4827 by 25, put two 
cyphers on the right, which is 
equivalent to multiplying by 100; 
and divide by 4. 
4)482700 

120675 Answer. 

To divide 582100 by 25, strike 
ofi^ two figures on the right hand, 
which is equivalent to dividing by 
1 00 ; then multiply by 4. 
5821 100 
4 



To multiply 6218 by 125, put 3 
cyphers, which is equivalent to 
multiplying by 1000; then di- 
vide by 8. 

8)6218000 



777250 Answer. 



To divide 4567000 by 125, strike 
off three figures on the right hand, 
which is equivalent to dividing by 
1000; then multiply by 8. 
45671000 
8 



23284 Answer. 



36536 Answer. 



2758 


2758 


3099 


3099 


4C9 




1029 


469 




1029 


7355 





CHAP. I.] VULGAR FRACTIONS. 13 

PROOF OF THE FIRST FOUR RULES OF ARITHMETIC. 

Simple as these four rules are, it is not unusual to commit errors 
in working them : it is, therefore, useful to possess modes of proof. 

1. Now, addition may be proved by 
adding downwards, as well as upwards, 
and observing whether the two sums 
agree ; or, by dividing the numbers, to be 
added into two portions, finding the sum 
of each, and then the sum of those two 
separate amounts. Thus, in the margin, ^ss^ 5857 

the sum of the four numbers is 7355 ; 

the sum of the two upper ones 5857, 1498 

of the two lower ones 1498, and their 

sum is 7355, the same as before. 7355 

2. The proof of subtraction is effected by adding the remainder to 
the subtrahend ; if their sum agrees with the minuend the work is 
right, otherwise not. 

3. Multiplication and division reciprocally prove each other. 
There is also another proof for multiplication, known technically 

by the phrase casting out the nines. Add together the numbers from 
left to right in the multiplicand, dropping 9 whenever the sum exceeds 
9, and carry on the rem£under, dropping the nines as often as the 
amount is beyond them ; and note the last remainder. Do the same 
i^-ith the multiplier and with the product ; then multiply the first two 
remainders and cast the nines out of their product; if the remainder 
is equal to the last remainder, this is regarded as a test that the work 
is right. Thus, taking the second example in multiplication, the 
figures in the multiplicand amount to 6 above two nines, those in the 
multiplier to 6 above three nines, those in the product to above six 
nines ; the product 6 x G of the two first excesses is 36, or above 
four nines : the coincidence of the two O's is the proof. It is plain, 
however, that the proof will be precisely the same so long as the 
figures in the product be the same, whatever be their order : the 
proof, therefore, though ingenious, is defective •. 
A similar proof applies to division. 



Sect. VI. Vulgar Fractions. 
The fractions of which we have already spoken in Sect. L, are 
usually denominated Common or Vulgar Fractions, to distinguish 

* The correctness of this proof, with the exception above specified, may be 
shown algebraicaUy, thus : — put M and JV — the number of nines in the mul- 
tiplicand and multiplier respectively, m and n their excesses ; then, 9 M •\- m =^ 
the multiplicand, and 9 Jv -f- n » the multiplier, and the product of those 
factors will l>e — 81 M N + 9 Af n + 9 iV m = m n ; but the three first 
terms are each a precise number of nines ; because one of the factors in each 
is so ; these, therefore, being neglected, there remains m n to be divided by nine ; 
but m n is the product of the two former excesses : therefore the truth of the 
method is evident. Q> E. D. 



14 VULGAR FRACTIONS. [PART I. 

them from another kind, hereafter to he mentioned, called DecinuU 
Fractions, 

A fraction is an expression for the value of any part of an integer, 
or whole numher, such numher heing considered as unity. Thus, if 
a pound sterling he the unit, then a shilling will he the twentieth 
part of that unit, and /our pence will he four twelfths of that twen- 
tieth part. These represented according to the usual notation of 
vulgar fractions, will he -^^ and -^-^ of -^^ respectively. 

The lower numher of a fraction thus represented (denoting the 
numher of parts into which the integer is supposed to he divided) is 
called the denominator ; and the upper figure (which indicates the 
numher of those parts expressed by the fraction) the numerator. 
Thus, in the fractions ^, -j^, 7 and 15 are denominators^ 5 and 8 
numerators. 

Vulgar fractions are divided into proper, improper, mixed, simple, 
compound, and complex. 

Proper fractions have their numerators less than their denomi- 
nators, as ^, ^, &c. 

Improper fractions have their numerators equal to, or greater than, 
their denominators, as ^, i^, &c. 

Mixed fractions, or numbers, are those compounded of whole 
numbers and fractions, as 7^9 12^, &c. 

Simple fractions are expressions for parts of whole numbers, as 
%, ^ &c. 

Compound fractions are expressions for the parts of given fractions, 
as I- of I, ^ of -3?^, &c. 

Cimijiex fractions have either one or both terms fractional 
52 12 6^ 
numbers, as ^^ j^, ^^ &c. 

The value of a fraction is not altered by multiplying or dividing 
both its numerator and denominator by the same number; thus 
i? \'> W^ Jli' *^® ^ equal, although successively multiplied by 2, 6, 
and 12. 

Any number which will divide two or more numbers without 
remainder, is called their common measure. 



REDUCTION OF VULGAR FRACTIONS. 

This consists principally in changing them into a more commodious 
form for the operations of addition, subtraction, &c. 

Case I. — To reduce fractions to their lowest terms, 

Rtde, — Divide the numerator and denominator of a fraction by any 
number that will divide them both, without a remainder ; the quotient 
again, if possible, by any other number : and so on, till 1 is the 
greatest divisor. 

Thus, m^ =z If* = ^\\ = i\ = I, where 5, 3, 7, 7, re- 
spectively, are the divisors. 

^"■5 IHi = h ^7 dividing at once by 735. 



CHAP. I.] VULGAR FRACTIONS. 15 

Nnfte, — This number 735 is called the greatest common measure of 
the terms of the fraction : it is found thus — Divide the greater of the 
two numbers by the less; the last divisor by the last remainder, and 
so, on till nothing remains : the last divisor is the greatest common 
measure required*. 

Case II. — To redu4X an improper f ration to its equivalent whole or 
mixed number. 

Rule, — Divide the numerator by the denominator, and the quotient 
will be the answer: as is evident from the nature of division. 

EtX, — Let ^^ and ^4^ be reduced to their equivalent whole or 
mixed numbers. 

43)957(22^} Answer. 274)5480(20 Answer. 

86 548 

97 

86 = 

11 

Cask III. — To reduce a mixed number to its equivalent improper 
Jraction ; or a whole number to an equivalent fraction having any 
assigned denominator. 

Rule, — This is, evidently, the reverse of Case II.; therefore multiply 
the whole number by the denominator of the fraction, and add the 
numerator to obtain the numerator of the fraction required. 

Ex, — Reduce 22|:^ to an improper fraction, and 20 to a fraction 
whose denominator shall be 274. 

(22 X 43) + 11 = 957 new numerator, and \^ the 1st 
fraction. 

20 X 274 z= 5480 new numerator, and ^^ the 2nd fraction. 

* The foUowing theorems are useful for abbreviating Vulgar Fractions :-— 
Theorems. 

1. If the last digit of any number be divisible by 2, the whole number is 
divisible by 2. If the two last dig:its be divisible by 4, the whole number is 
divisible by 4. If the three last digiu be divisible by 8, the whole number is 
divisible by 8. And, generally, if the last n digits of any number be divisible by 
2*, the whole number is divisible by 2*^. 

2. If a number terminate with 5, it is divisible by 5 ; and if it terminate in 
0, it is divisible by either 10 or o. 

3. If the sum of the digits constituting any number be divisible by 3 or 9, the 
whole is divisible by 3 or 9 ; and if also the last digit be even, the whole number 
is divisible by 18. 

4. If the sum of the digits constituting any number be divisible by 6, and the 
right.hand digit by 2, the whole is divisible by 6 : for by the data it is divisible 
both by 2 and 3. 

5. If the sum of the 1st, 3rd, 5th, &c , digits constituting any number be equal 
to that of the 2nd, 4th, 6th, Ac., that number is divisible by 11 : for if a, 6, c, 
(f , e, m, n, be the digits, constituting any number, its digits, when multiplied 
by 11, will become 

(8) (7) (6) (5) (4) (3) (2) (1) 

«, a+4, A+c, c+rf, rf-h^, c-fm, m-^n^ n ; 

where the odd temns are » to the even. 



id VULGAR FRACTIONS. [pART I. 

Case IV. — To reduce a compound frartion to an equivalent simple 
one. 

Rule, — Multiply all the numerators together for the numerator, and 
all the denominators together for the denominator, of the simple 
fraction required. 

If part of the compound fraction be a mixed or a whole number, 
reduce the former to an improper fraction, and make the latter a 
fraction bv placing 1 under the numerator. 

When like factors are found in the numerators and denominators, 
cancel them both. 

Ex,— Reduce |? of \ of ^ of ^ of -,8^ to a simple fraction. 

2x3xr> x7x8 2x5x8 1x6x8 1x5x4 20 

3X4X7X9 Xll""-! X 9 X ll""2 X 9 X ll""! X 9 X ll^OQ 
Here the 3 nnd 7 common to numerator and denominator are first 
cancelled ; then the fraction is divided by 2 ; and then by 2 again. 

/t^x.— Reduce throe farthings to the fraction of a pound sterling. 

A farthing is the fourth of a penny, a penny the twelfth of a shilling, 
nnd a shilling the twentieth of a pound. 

Therefore ^ of -f^y of ^ zz ^J^ = -^^^y the answer. 

2^ 
Ex, — Simplify the complex fraction ... 

H 

Here, reducing the mixed numbers to improper fractions, we have 

8 

— : multiplying by 3, to get quit of the denominator of the upper 
V 

8 
fraction, we have p: multiplying by 5, to get quit of the denominator 
of the lower fraction, we have ^J: dividing both terms of this fraction 
by 8, there results ^ for the simple fraction required. 

Case V. — To reduce fraction% of different detioininators to equi- 
valent fractions having a common detiominaior. 

Rule. — Multiply each numerator into all the denominators except 
its own, for new numerators ; and all the denominators together for a 
common denominator. 

Ex, — Reduce |, ^, and §, to equivalent fractions having a common 
denominator. 

2 X 7 X 9 = 126 \ 

6 X 3 X 9 = 162 [ the numerators. 
5 X 3 X 7 = 105 ) 

3 X 7 X 9 = 189, the common denominator. 
Hence the fractions are -[|^, -^, ^-g^, or *«, * J, ^^, when divided 
by 3. 

Hence, also, it appears that f exceed §, and that 4 exceed ^, 

Ex, — Reduce | of a penny, and J of a shilling, each to the fraction 
of a pound ; and then reduce the two to fractions having a common 
denominator. 

* of a penny = ^ of ^^ of 4^ = -p*,^^ = ^^^ of a pound. 

% of a shilling = ^ of ^^ = ^% = ^ = ^ of a pound. 

Hence ^ of a shilling are 1 times as much as | of a penny. 



CBiP. I.] VULOAB FRACTIONS. 17 

Mute.— Other methods of reduction will occur to the student after 
tolenble practice, and still more after the principles of algebra are 
ttqaired. 

ADDITION AND SUBTRACTION OP FRACTIONS. 

RuU,—lf the fractions have a common denominator, add or sub- 
tract the numerators, and place the sum or difference as a new numc- 
ntor oTer the common denominator. 

If the fractions have not a common denominator, they must be re- 
duced to that state before the operation is performed. 

In addition of mixed numbers, it is usually best to take the sum of 
tbemtegere, and that of the fractions, separately ; and then their sum, 
for the resolt required. 

Examples. 

1. Find the sum of f , ^, and |. 

. l;f- ^ t* "T-P + H + fi = V»' = 2H- 

z. Take | of a shilhng from y^ of a pound sterhng. 

I of a shilling = | of ^>„ = ^^ of a pound = -^q. 

Also -rV of a pound = /^j. Hence ^% - -^^^ = ^^Vff 

= ^»j = 11 pence. 
3. Find the difference between 12 J and 8 J. 

»2| -H = V - V = W - Vo" = SV = 4/»- 

MULTIPLICATION AND DIVISION OF FRACTIONS. 

R*ik 1. To multiply a fraction by a whole number, mtdtiplif the 
KMPKrttfor by that number, and retain the denominator : — Or, divide 
^dtnomnator by the same number, (if a multiple of it,) and retain 
tbe numerator. 

2. To divide a fraction by a whole number, multiply the denomi- 
■•tor by that number, and retain the numerator : — Or, divide the 
MJWrotor by the same number, (if a multiple of it,) and retain the 
Nominator. 

3. To multiply two or more fractions is the same as to take a 
friction of a fraction ; and is, therefore, effected by taking the pro- 
duct of the numerators for a new numerator, and of the denomina- 
tors for a new denominator. (The product is evidently smaller tlian 
eitber factor when each is less than unity.) 

*. To divide one fraction by another, invert the divisor, and pro- 
«wd as in moltiplication. (The quotient is always greater than the 
dividend when the divisor is less than unity.) 

Examples. 
1. Multiply { by 2, and divide ^ hy 5. 

c 



18 



DECIMALS. 



[PABT I. 



Multiply 25 by f , and divide f by ji 
1-i, Ans, 



3 =8 ^2;andf --^=^ 



l-V = 



Afultiply £2 IBs. U. bv 3}, and divide £4 15«. by 3^, 
£2 13«. 4</. = 2 + IJ + T*/ of A = 21 = f, and 



= 8 X J = V == «/ = 9^ = £9 6«. 8</. 

£4 158. H- 3i = 4| -r- 3i = V -^ V = V 
liJ = £1 8«. 6</. 
i^Tofe. — In the multiplication of mixed 
numbers, it is often less laborious to 



iV 






perform the multiplication of each part 
separately, and collect their sum, as in 
the margin, than to reduce the mixed 
numbers to improper fractions, and re- 
duce their product back again to a mixed 
number. 



45 
45 

8 
T 

45 
17 



Multiply 45f 
ByJTf 
7 =315 
1 ten 3= 45 . 

I = .30 

% = 'jn 

Product 808^ 



Sect. VII. Decimal Fractions, 



The embarrassment and loss of time occasioned by the com- 
putation of quantities expressed in vulgar or ordinary fractions, have 
inspired the idea of fixing the denominator so as to know what it is 
without actually expressing it. Hence originate two dispositions of 
numbers, decimal fractions and complex numbers. Of the latter, 
such, for example, as when we express lineal measures in yards, in 
feet (or thirds of a yard), and inches (or twelfths of a foot), we shall 
treat in the following section, and shall here confine ourselves to the 
former. 

Decimal fractions, or substantively, decimals^ are fractions always 
having some power of ten for their denominator ; but for the sake of 
brevity only the numerator is expressed, being written as tL^whok 
number with a dot placed on its left hand, which dot determines the 
value of the denominator, the number of cyphers in the denominator 
being always equal to the number of figures to the right of the dot, 
or as it is termed decimal point ; if the number of significant figures 
in the numerator is not sufficient, cyphers are added to the left hand. 
It is evident that the values of decimals decrease in the same tenfold 
proportion from the point to^^-ards the right hand, as those of integers 
mcrease towards the left : — thus 



Igai 


r fraction, 


tV is written 


•1 


>i 


99 


aha 99 


•01 


» 


99 


TO (TU 99 


•001 


>i 


99 


TTJOcJi) 99 


•0001 


»> 


99 


iV >9 


•7 


» 


99 


tVo »> 


•43 


» 


99 


"rioinr »> 


•0125 



i« 


written 


•3 


* 


>i 


•6 


4 


y» 


•428571 


m 


>f 


•29504 



CHAP. I.] DECIMALS. 19 

The vulgar fraction, 7-/^ is written 7-3 
42^*5 „ 42-85 

57,VoV >> 57-217 

&c. &c. 

The Talae of a decimal fraction is not altered by cyphers on the 
Jight hand : for '500, or -^W^, is in value the same as ^^, or -5, 
that is |. 

When decimals terminate after a certain number of figures, they are 

called/iii^«,as-125 = Vt5V(jpi-958 = T^V« = m- ^ . . 
When one or more figures in the decimal become repeated, it is 
oaHed a repeating or circulating decimal ; and a dot is placed over the 
figure to be repeated, if only one, or if more than one, over the first 
and last figures ; thus : — 

•333333, &c. 

'666666, &c. 

•428571428571, &c. 

•29504504, &c. 

When the circulating portion of the decimal is preceded by other 
figures which do not circulate, (as in the last example,) it is called a 
mixed circulate. 

Rules for the management of this latter kind of decimals are given 
by several authors ; but, in general, it is more simple and commodious 
to perform the requisite operations by means of the equivalent 
vulgar fractions, the method of obtaining which is given in Case III. 

RBDUCTION OP DECIMALS. 

Reduction of Decimals is a rule by which the known parts of given 
integers are converted into equivalent decimals, and vice verad. 

Case I. — To reduce a given vulgar fraction to an equivalent 
dedmoL 

Rule. — Annex as many cyphers to the numerator as may be neces- 
sury, then divide by the denominator, and point off in the quotient as 
many places of decimals as the number of cyphers added to the 
numerator; if the quotient does not contain so many figures, the 
deficiency must be made up by cyphers placed on the left hand. 

JSxamplea. 
1. Reduce ^, |, i^, ^, to equivalent decimals. 
2) 1-0 4) 3-00 

•5 decimal = | ; *75 decimal = ^ : 

C 4)70000 ( 8)6000 

16} «4 

^ 4)1-7500 I 8)^75000 

•4375 decimal =- -f^; '09375 decimal = ^ 

•asHB - c 2 



20 DECIMALS. [PABT I. 

2. Reduce ^\ and ^i to equivalent decimals. 
/ 3)4-000000 
27 



( 9)1*333333 



•148148, &c. = 



•148 decimal z= ^\ 



i 7)11-0000000 
63' 



I 9) 15714285714285 

•1746031746031, &c = 



•174603* decimal = i^. 

These two are evidently circulating decimals, in the former of 
which the figures 148 become indefinitely repeated, in the latter the 
figures 174603. 

3. Reduce 14«. 6d, to the decimal of a pound. 

First, 14*. ed. = JJ + i of ^ig = IS + ViJ = 18- 
Then J 8 — 1^" = ''^^^' '^® decimal required. 

4. Reduce tJ^ to its equivalent decimal. 

57) 44-000000 (-77192, &c. decimal = ^. 
399 




170 
114 

56 

Note, — The above fraction is = J x |^, of which the two denomi- 
nators arc both jortm^ numbers^ (that is, divisible by no other number 
than unity,) the entire equivalent decimal is a circulate of 18 places, 
t. e. one less than the last prime .... -771929824561403508, 7719, 
&c. over again ad infinitum*, 

• There are many curioui properties of fractions i « 'i 42867, Ac 

whose denominators are prime niimliers, one of I ^ •285714 An 

which may lie here shown in reference to fractions * "* . ' .» ^' 

having the denominator 7- The circulating fi^nires ? ~ •428671, Ac 

of the equivalent decimals are precisely the same, ^ "" -671428, Ac 

for \y ), &C., and in the same order: the cir- & ^ *714286 Ac 

culate merely oommeooes at a different place for each Z a.. , .<:* . 

numerator. * * 867142, Ac 



CHIP. I.] DECIMALS. 21 

Gin II. — Antf decimal being given to find its equivalent vulgar 
fnttm; or to express its value bg integers of lower denominafions, 

i^.— When the equivalent vdgar fraction is required, place 
Qoder the decimal as a denominator a unit with as many cyphers as 
there are figures in the proposed decimal ; and let the fraction so con- 
ititnted be reduced to its lowest terms. 

Or, if the value of the decimal he required in lower denominations, 
Doltiplj the given decimal hy the number of parts in the next less 
deDomioation contained in its integer; and point off, from right to 
left, aa many figures of the product as there were places in the given 
decimal. Multiply the decimal last pointed off by the value of its 
integer, in the next inferior order, pointing off the same number of 
dednuJs as before : and thus continue the process to the lowest 
bteger, or until the decimals cut off become all cyphers ; then will 
tkeseTenJ numbers on the left of the separating points, together with 
the remaining decimal, if any, express the required value of the given 
decimal. 

Examples, 

1. Find the vulgar fractions equivalent to '25 and '375. 
•25 = ^V^ = i ; and '375 = ^Vrf^^ = |, Answers. 

2. Fbd the value in shillings, &c. of '528125 of a pound. 

•528125 
20 



10-562500 >Ans. lOs. Bid 

12 \ 



6-7500 = 6| 



3. Fmd the value of *74375 of an acre. 
•74375 

4 



2-97500 > Ans. 2 roods 39 perches. 
40 ^ 



39000 



Cm III. — To reduce a circulating decimal to its equivalent vtdgar 
fi^ttum, 

BtJs, — Take the figures in the decimal and place them as a whole 
Biimber for the numerator, and under them for a denominator as many 
S*! aa there are figures in the circulate, and the fraction thus formed 
will be equivalent in value to the given decimal. 

If the decimal is a mixed drcukte, subtract the finite part (or the 
%iii«s which are not repeated) from the whole mixed circulate, (both 



22 DECIMALS. [part I. 

considered as whole numbers,) for the numerator; and for the deno- 
minator, take as many 9's as there arc figures in the circulating por- 
tion of the decimal, with a8 many cyphers to the right as there are 
figures in the finite portion of the same. 





Examples. 




The circulate 3 


= i 


i 


•06 


= ^ = 


h 


» -549 


— wJ *^ 


^ 


» 7630 


= "rm = 


7A 


The wwerf circulate -6409 = ^^ll^o* 


= il* 


5-29504= 5»^|J-J^«5' = 5iH 



ADDITION AND SUBTRACTION OF DICIMALS. 

These operations are performed precisely as in whole numbers, the 
figures being so arranged that units stand under units, tens under tens, 
&c., or, ^which amounts to the same thing,) so that the decimal points 
stand under one another. Thus, 



421-75 From 24861 78 



Add } 32-8165 Take 14-56789 



together i -0027 



11- Remains 2471*60511 



Sum 465-5692 Proof 2486-17300 



MULTIPLICATION AND DIVISION OP DECIMALS. 

Here, again, the operations are performed as in integers : Then, in 
multiplication, let the product contain as many decimal places as 
there are in both the multiplier and multiplicand, cyphers being pre- 
fixed, if necessary, to make that number ; and, in division, point off 
as many decimals in the quotient as the number in the dividend 
^including the cyphers supplied, if there be any) exceeds that in the 
oivisor. 



Examples. 
Multiply 43-7 by 39 1, and 2 4542 by -0053. 



43-7^ 
391 


' ^Here 437 x 391 

^^-}88S^ = 170^*0%. 

as in the decimal 
i operation. 


2-4542 
•0053 


437 
3933 . 
1311 


73626 
122710 




•01300726 


170-867 





Here one cypher 
is prefixed to make 
the requisite num- 
ber of decimals in 
the product. 



CBIP. I.] COMPLEX NUMBBBS. 23 

2. Divide 172*8 by -144, and 192 by 5*423. 

•144) 172-8 ( 1200- quotient. 5423 ) 192000 ( 35-40475 

144 16269 



288 29310 

288 27115 



00 21950 

= 21692 



In the first of these examples, the two 25800 

cjphen brought down, together with the deci- 21692 

mil 8, make the number of decimals in the 

diridend the same as in the divisor, there- 41080 

fore the quotient is composed entirely of in- 37961 

tegers. In the second example, 3, the num- 

ber of decimal places in the divisor, taken 31190 

from 8, the nnmber in the dividend (including 271 15 

thoK brought down), leave 5 for the decimtd 

places in the qnotient. 4075 



Skt. VIII. Complex Fractions used in the Arts and Commerce. 

In the arts and in commerce, it is customary to assume a series of 
nnits bsTing a constant relation to each other, so that the units of 
ODe denomination become fractions of another. One farthing, for 
eumple, is ^ of a penny, 1 penny -/.t of a shilling, 1 shilling ^*jy of a 
poand, or ^y of a guinea. One lineal inch, again, is yV of a foot, 1 foot 
i of A yard ; and so on, according to the relations expressed in the 
tible« in Chap. V. on Mensuration, Sect. I. The arithmetical opera- 
tioDs on complex numbers of these kinds are usually effected by 
limpler rules than those which apply to vulgar fractions generally ; 
of which it will, therefore, be proper here to specify a few. 

BBDUCTION. 

Here we have two general cases : 

Ca81 I. — When the numbers are to be reduced from a higher de- 
nomination to a lower : — 

Bide. — Multiply the number in the higher denomination by as 
■uy of the next lower as make an integer, or one, in that higher, 
<id set down the product. To this product add the number, if any, 
vhich was in this lower denomination before ; and multiply the sum 
hy as Buny of the next lower denomination as make an integer in 
tht present one. Proceed in the same manner through all the dcnomi- 
B*^s to the lowest, and the number last found will be the value 



24 



COMPLEX NUMBERS. 



[part I. 



of all tlic numbers whicli were in the higher denominations taken 
together. 

Case IL — fV/ten the numbers are to be reduced from a lower de- 
nomination to a higher : — 

Rule.—DWidc the given number by as many of that denomination 
as make one of the next higher, and set down what remains. Divide 
the quotient by as many of this as make one of the next higher de- 
nomination, and set down what remains in like manner as before. 
Proceed in the same manner through all the denominations to the 
highest; and the Quotient last found, togetlier with the several re- 
mainders, if any, will be of the same value as the first number proposed. 

The method of proof is to work the question back again. 

Examples, 

1. Reduce <£l4 to shillings, pence, and farthings; and 24316 
farthings into pounds, &c. 



14 


4)24316 


20 

280 shillings 
12 




12) 6079 pence 
20) 506 7 


3360 pence 
4 


£25 6«. 7d. 


13440 farthings 





2. Reduce 22 Ac. 3 R. 24 P. into perches; and 52187 perches 
into acres. 



a. r. p. 
22 3 24 


40)52187 


4 




91 roods 


4)1304 27 


40 


Ac. 326 R. 27 P. 


3664 perches. 






ADDITION. 



/2w/e.— Place the quantities to be added so that those of the same 
denomination may be all under each other. Then add up the numbers 
in the right hand column, and divide their sum by the number of units 
of that denomination contained in an unit of the next denomination 
to the left, write the remainder (if any). at the foot of the first column, 
and carry the quotient on to the addition of the second column^ and 
thus proceed until all have been added up. 



CHIP. I.] COMPLEX NUMBSBS. 25 

Examj^es. 

£ «. dL lb. oar. duft. gr, tb, om. dujt.gr, 

'368 10 3 / 14 6 12 13 / 10 8 11 17 

1257 10 5 i 17 5 3 12 t 42 5 16 12 

.jJ 88 U 4i . ,, J 15 9 16 . , J 12 2 14 18 

^^^j 33 10 ^"^"^i 2 7 15 20 ^"^"^ \ 51 6 22 

f 12 13 5 f 13 2 10 19 f 24 9 17 17 

8 8 8^ \ 4 1 5 21 V 29 4 18 22 



Sum 769 4 2 Sam Q^ 11 18 5 Sum 171 2 12 



SUBTRACTION. 

Ruk, — Write the smaller number beneath the greater, taking care 
to keep the same denominations under each other. Then begin at the 
right band and subtract the lower number of each denomination from 
the upper, writing the remainder underneath. 

When the lower number of any denomination is greater than the 
upper one of the same, add to the latter as many units as are con- 
tamed in one unit of the next greater denomination, always taking 
care, when such has been done, to add one to the next lower number 
to the left. 

Examples, 

£. M. d. £. g, d, lb, OM,dtoLgr. 

From 16 12 8f From 21 13 4J From 18 9 10 8 

Take 10 11 sl Take 18 9 8| Take 9 10 15 20 



Bern. 6 1 2| Rem. 3 3 8| Rem. 8 10 14 12 



MULTIPLICATION. 



i^ti/e.— Place the multiplier under the lowest denomination of the 
multiplicand. --Multiply the number in the lowest denomination by 
the moltiplier, and find how many integers of the next higher de- 
nomiDation are contained in the product, and write down what 
remains. — Carry the integers, thus found, to the produce of the next 
higher denomination, with which proceed as before ; and so on, 
thioagh all the denominations to the highest ; and this product, 
together with the several remainders, taken as one number, will be 
the whole amount required. 

If the multiplier exceed 12, multiply successively by its component 
puis; as in the following examples :•— 



26 OOMPLEX NUMBKBS. [PABT I;. 

ExamfleM, 

£ 9m d, a. r. p. 

1. Mo]tiplj4 17 e\ by 441, and 3 2 14 by 531. 

10 

£ $, d. 

4 17 Q\ 85 3 20 

9 X 7 X 7 = 441 9 10 



43 


17 


lOj 

7 


307 


4 


7 


An*. £2150 


U 


lOj 



358 3 for 100 
5 



1793 3 for 500 
107 2 20 3 times 10 
3 2 14 1 top line. 



^n«. 1904 3 34 



DIVISION. 

Bide. — Place tbe divisor and dividend as in simple division. — Begin 
at the left hand, or highest denomination of the dividend, which divide 
by the divisor, and write down the quotient. — If there be any re- 
mainder after this division, find how many integers of the next lower 
denomination it is equal to, and add them to the number, if any, 
which stands in that denomination. — Divide this number, so found, 
by the divisor, and write the quotient under its proper denomination. 
— Proceed in the same manner through all the denominations to 
the lowest, and the whole quotient, thus found, will be the answer 
required. 

£ B. d. 

2. Divide 521 18 6 by 432. 

432 = 12 X 12 X 3. 



Therefore, by short division : 
12)521 18 6 



12)43 9 10^ 



3)3 12 6i + I a farthing. 



Qtwiient £} 4 1| + f of a farthing. 



CHAP. I. 



DUODECIMALS. 



27 



By loDg division : — 

£ 9. d. £ 9. d. 
432)521 18 6 (1 4 1 J + f of a farthing. 
432 

89 
20 

432)1798(4 
1728 

70 
12 

432)846(1 
432 

414 

4 



432)1656(3 
1296 



m^a^i 



360 



DUODECIMALS. 

Fmetions whose denominators are multiples of 12, as 144, 1728, 
&e., ire called duodecimals; and the division and suh-di vision of the 
iateger sre Hudersiood without being expressed, as in decimals. The 
method of operating by this class of fractions is principally in use 
>DiODg artificers, in computing the contents of work, of which the 
^eDsions are taken in/eet, inches^ and twelfths of an inch. 

Rule. — Set down the two dimensions to be multiplied together, 
one under the other, so that feet shall stand under feet, inches under 
iocbes, &c. Multiply each term in the multiplicand, beginning at the 
West, by the feet in the multiplier, and set the result of each 
iou&ediately under its corresponding term, observing to carry 1 for 
vvery 12, from the inches to the feet. In like manner, multiply all 
tbe multiplicand by the inches of the multiplier, and then by the 
twelfth parts, setting the result of each term one place removed to 
tbe right hand when the multiplier is inches, and two places when the 
parts become the multiplier. The sum of these successive products 
will be the answer reqmred. 

Or, instead of multiplying by the inches, &c., take such parts of 
the multiplicand as these are of a foot. 



28 



P0WBB8 AND ROOTS. 



[PABT 1. 



Examples. 

1. Multiply \2ft. 7i ins. by 1 /t. 3 J tVw. 

//. in$. ' 

12 7 4 or, 

7 3 9 



88 3 4 

3 1 10 

9 5 6 

92 2 7 6 



tnM. 

3 == i of 1 ft. 
9' = I of 3 ins. 



fL %n», ' 
12 7 4 

7 ft. 



88 3 4 
3 I 10 
9 5^ 



92 2 7| 



2. Multiply Z5fi. ^\ins. into 12^. S^ins. 

ft. ins. ' 

35 4 6 or, 

12 3 4 



tnM. 

3 = ^ of 1 ft. 
4' = ^ of 3 ins. 



424 6 

8 10 1 

11 9 


6 
6 


434 3 11 






ft. ins. ' 
35 4 6 

12 ft. 



424 


6 





8 


10 


n 




11 


9i 


434 


3 


_n 



The feet in the answers are square feet, but the numbers standing 
in the place of inches are not square inches but twelfth parts of 
square feet, each part being equal to 1 2 square inches ; and the 
numbers in the third place being twelfth parts of these are square 
inches : in like manner, if the operation be carried further, every 
successive place will be a twelfth part of that preceding it. 



Sect. IX. Powers and Roots. 

• A power is a quantity produced by mttltipl}'ing any given number, 
called the root or radix, a certain number of times successively by 
itself. The operation of thus raising powers is called involution. 
Thus, if 3 is the root, 

3 = 3 is the Ist power of 3. 
3x3=3'= 9, is the 2d power, or square of 3. 
3 X 3 X 3 = 3 '= 27, is the 3d power, or cube of 3. 
3 X 3 X 3 X 3 = 3^ = 81, do. 4th power, or biquadraU of 3, 
&c. &c. &c. 



CHAP. I.] 



BVOLUTION. 



29 



L- ,., 



Table of thejirsi Nine Patters of the first Nine Numbers. 



lit 

A 


2d 
1 
4 

9 
16 


3d 

1 
8 


4di 


&th 


6tk 


7tli 


&th 


dUi 




1 


1 


1 


1 


1 


1 




16 
81 


32 


64 


128 


356 


61t 




243 


729 


2187 


6fi61 
65536 


19683 




64 


G2& 
1296 


1024 
3125 


4096 
1S625 


16384 


£^144 




3S 1 1S5 


78126 
279936 


390625 
1679616 


1953125 




m 

64 


216 


7776 


4G6&G 


10077696 




343 240] 
512 4006 


16B07 


117649 


823543 
2097152 


5764801 

— ^ — 

16777216 


4035364^7 




32768 


262144 


134317728 






tl 


729 6561 


^9049 


531441 


47629C^ 


43046731 


367420480 





So again, J x f = $ = square off;|x| = ^ = cube of 
3 > ^ X J- = ^, biqoadrate of 4 ; and so of others. Where it is 
eyident, that while the powers of integers become successively larger 
^nd larger, the powers of pure or proper fractions become sue- 
oessively smaller and smaller. 



EVOLUTION. 

Evdulumy or the extraction of roots, is the reverse of involution. 

Any power of a given number may be found exactly ; but we 
cannot, conversely, find every root of a given number exactly*. 
Thus, we know the square root of 4 exactly, being 2 ; but we cannot 
assign exactly the cube root of 4. So, again, though we know the 
cttbe root of 8, viz. 2, we cannot exactly assign the square root of 8. 
But, of 64 we can assign both the square root and the cube root, the 
former being 8, the latter 4. 

By means of decimals we can in all cases approximate to the root 
to any proposed degree of exactness. 

Those roots which only approximate are called surd roots, or surdSy 
or irrational numbers ; as v^2, V5, \/9> &c., while those which can 
be found exactly are called rational; asx/9 = 3, i/125 = 5, 
iyie == 2. 

1. — To extract the square root. 

Rule, — Divide the given number into periods of two figures each, 
by setting a point avei' the place of unitSy another over the place of 

* For the method of extracting roots by logarithm!, see page 47* 



dO EVOLUTION. [pari I. 

hundreds, and so on over every second figure, both to the left hand in 
integers, and to the right hand in decimals. Find the greatest square 
in the first period on the left hand, and set its root on the right hand 
of the given number, after the manner of a quotient figure in division. 
Subtract the square thus found from the said period, and to the 
remainder annex the two figures of the next following period, for a 
dividend. Double the root above-mentioned for a divisor ; and find 
how often it is contained in the said dividend, exclusive of its right- 
hand figure ; and set that quotient figure both in the quotient aad 
divisor. Multiply the whole augmented divisor by this last quotient 
figure, and subtract the product from the said dividend, bringing down 
to it the next period of the given number, for a new dividend. 
Repeat the same process, viz., find another new divisor, by doubling 
all the figures now found in the root; from which, and the last 
dividend, find the next figure of the root as before; and so on through 
all the periods, to the last*. 

Note. — The best way of doubling the root, to form the new 
divisors, is by adding the last figure always to the last divisor, as ap- 
pears in the following Examples. — Also, after the figures belonging to 
the given number are all exhausted, the operation may be continued 
into decimals at pleasure, by adding any number of periods of cyphers, 
two in each period. 

Examples. 

1. Find the square root of 17*3056. 

17*3056(4'16 the root : in which the number of 
IG decimal places is the same as the 

number of decimal periods into which 

the given number was divided. 



81 


130 


1 


81 


826 


4956 


6 


4956 



* The reason for separating the figures of the dividend into periods or 
portions of two places each, is, that the square of any single figure never con- 
sists of more than two places ; the square of a number of two figures of not 
more than four places, and so on. So that there will be as many figures in the 
root as the g^iven number contains periods so divided or parted off. 

And the reason of the several steps in the operation appears from the algebraic 
form of the square of any number of terms, whether two or three, or more. 
Thus, 36* » 30* + 2 . 30 . 5 + 6% or genmilly (a -f 6)* = a* + 2 a 4 -|- i» = 
a* 4* (2 a 4* 6) ^t the square of two terms ; where it appears that a is the first 
term of the root, and b the second term ; also a the first divisor, and the new 
divisor is 2 a 4* 6, or double the first term increased by the second. And hence 
the manner of extraction is as in the rule. 



CHAP. I.] 



EVOLUTION. 



31 



2. !Rnd the square root of 2, 
to six decimals. 

2(1-414213 root. 
1 



24 1 

4 


00 
96 


281 

1 


400 
281 


2824 

4 


11900 
11296 


28^82 
2 


60400 
56564, 


28284 


1 383600 
1 282841 



2828423 



10075900 
8485269 

1590631 



3. Find the square root of -f-^. 
-r\ = '4>16666666y &c. 

6-4i 6666(0-64549, &c. 
36 



124 566 
4 496 


1285 
5 


7066 
6425 


12904 64166 
4 51616 


12908 


9 


1255066 
1161801 



93265 



^ote. — In cases where the square roots of all the integers up to 
1000 are tabulated, such an example as the above may be done more 
easily by a little reduction. ThusV^j =\/(t^^ X ^f) =x/ -j^^ = 

^^60 = ?:?^ = -645497, &c. 

2. — To extract cube and higher rooU. 

The rules usually given in books of arithmetic for the cube and 
higher roots, are very tedious in practice: on which account it is 
advisable to work either by means of approximating rules, or by 
means of logarithms*. The latter is, generally speaking, the best 
method. We shall merely present here Dr. Hutton's approximating 
rale for the cube root. 

Rule, — By trials take the nearest rational cube to the given num- 
ber, whether it be greater or less, and call it the assumed cube. 

Then say, by the Rule of Three, as the sum of the given number 
and double the assumed cube, is to the sum of the assumed cube and 
double the given number, so is the root of the assumed cube, to the 
root required, nearly. Or, as the first sum is to the difference of the 
given and assumed cube, so is the assumed root, to the difference of 
the roots, nearly. 

• See page 47- 



32 PROPOBTION. [part I. 

Again, by using, in like manner, the cube of tbe root last found 
as a new assumed cube, another root will be obtained still nearer. 
And so on as far as we please ; using always the cube of the last 
found root, for the assumed cube. 

Example, 
To find the cube root of 210358. 
Here we soon find that the root lies between 20 and 30, and then 
between 27 and 28. Taking therefore 27, its cube is 19683, which 
is the assumed cube. Then 



19683 
2 


21035-8 
2 


39366 
21035-8 


42071-6 
19683 


As 60401-8 


: 61754-6 
27 




4322822 
1235092 



27 : 27-6047 



60401-8)1667374-2(27-6047 the root nearly. 

459338 Again, assuming 27*6 

36525 and working as before, the 
284 root will be found to be 
42 27-60491. 



Sect. X. Proportion. 

Two magnitudes may be compared under two different points of 
view, that is to say, either by inquiring what is the excess of one above 
the other, or hoto often one is contained in the other. The result of 
this comparison is obtained by subtraction in the first case, by division 
in the second, the quotient resulting being termed the ratio of the 
two numbers. Thus 3 maybe regarded as the ratio of 12 to 4, since 
*3p or 3 is the quotient of the numbers 12 and 4. 

The first of two numbers constituting a ratio is called the antece- 
dent^ the second the consequent. 

The difference of two numbers is not changed by adding one and 
the same number to each, or by subtracting the same number from 
each. 

Thus 12 - 5 = (12 + 2) - (5 + 2) = 14 - 7 = (12 - 2) - 
(5 - 2) = 10 — 3. 

In like manner, a ra^io is not changed by either muUiplging both 
its terms, or dividing both its terms by the same number. 

Thus V =(y X !) = !« =c/-f)=f 



CHIP. I.] PROPOBTION. 33 

F/lnaliiy of differences^ or equidifference^ is a term used to indicate 
tlattbeMerence between two numbers is the same as the difference 
between two other numbers. Such, for example, asl2 — 9 = 8 — 

£gva/i<jf of ratios^ or proportion^ is similarly employed to denote 

that tbe ratio of t^wo numbers is the same as that between two others. 

TbusSO and 10, 14 and 7, have 2 for the measure of the ratio : we 

kie therefore a proportion between 20 and 10, 14 and 7, which is 

tbos expressed, 20 : 10 : : 14 : 7, and thus read 20 are to 10 a« 14 

are to 7. The same proportion may also be represented thus, ^g = y . 

Tboagb, \>y whatever notation it be represented, it is best to read or 

eoamer&te it as above. It is true, however, that in all cases when 

tvo fractions are equal, the numerator of one of them is to its de- 

QOffiinator, as the numerator of the other is to its denominator. 

In a proportion, as 20 : 10 : : 14 : 7, the second and third terms 
we called the meanSy the first and fourth the extremes. 

Wben the two means are equal, the proportion is said to be con- 
tinued. Thus 3 : 6 : : 6 : 1 2 are in continued proportion. This is 
osually expressed thus -rr 3 : 6 : 1 2 ; and the second term is called 
the mean proportional. 

In the case of equidifference^ as 1 2 — 9 =7 — 4, the sum of the 
extremes (12 + 4) is equal to that of the means (9 -r 7). In like 
JMnnerin a proportion, as 20 : 10 :: 14 : 7, the product of the ex- 
tremes (20 X 7) is equal to that of the means (10 x 14). The 
converse of this likewise obtains, that if 20 x 7 = 10 X 14, then 
20:10:: 14 : 7. Hence, 

1. If there be four numbers, 5, 3, 15, 9, such that the products 
5 X 9 and 3 x 15 are found equal, we may infer the equality of 
tbcir ratios, or the proportion ^ = ^ , or 5 : 3 : : 15 : 9. So that a 
proportion may always be constituted with the factors of two equal 
products. 

2. If the means are equal, their product becomes a square ; there- 
fore the mean proportional between two numbers is equal to the square 
root of their product. Thus, between 4 and 9 the mean proportional 
i8%/(4 X 9) = 6. 

3. If a proportion contain an unknown term, such, for example, as 
5:3:: 15: the unknown quantity; since 5 times the unknown 
<IttDtity roust be equal to 3 x 15 or 45, that quantity itself is equal 
to 43 -r 5 or 9. Or generally, one of the extremes is equal to the 
prodact of the means divided by the other extreme ; and one of the 
OKans is equal to the product of the extremes divided by the other mean. 

4. We may, without affecting the correctness of a proportion, 
object the several terms which compose it to all the changes which 
ciQ be made, while the product of the extremes remains equal to that 
ofthemeans. Thu8,for5 : 3 : : 15 : 9, which gives 5 x 9 = 3 x 15, 
we may 

I. Change the places of the means without changing those of the 
^^Etremea, or change the places of the extremes without changing those 
of the means : this is denoted by the term aUernando, 

D 



84 BULK OF THREE. [^A&T I. 

Thus, 5 : 3 : : 15 : 9 

become 5 : 15 : : 3:9 

or 9 : 3 : : 15 : 5 

or 9 : 15 :: 3 : 5 

II. Put the extremes in the places of the means ; this is oalled 
invertendo ; as 

3 : 5 : : 9 : 15 

III. Multiply or divide the two antecedents or the two consequents 
by the same number. 

It also appears, with regard to proportions, that the sum or the dif- 
ference of the antecedents is to that of the consequents, as either ante- 
cedent is to its consequent. 

And, that the sum of the antecedents is to their difference, as the 
sum of the consequents is to their difference. 

Tu ^ 3; 15 , ,, , 5 -h 15 5^15 

^^"^ "3"T^ = 1^ = v> and ^-^ —rz^' 

If there be a series of equal ratios represented by ^ = ijp = y == 

6 ^ 10 + 14 -f 30 

3g, we shall have 3 ^ ^ ^ ^ ^. ^^ = JJ = | = V> = &c. 

Therefore, in a series of equal ratios, the sum of the antecedents is to 
the sum of the consequents, as any one antecedent is to its consequent. 

If there be two proportions, as 30 : 15 : : 6 : 3, and 2 : 3 : : 4 : 6, 
then multiplying them term by term, we shall have 30 x 2:15 x 
3 : : 6 x 4:3 x 6, which is evidently a proportion, because 30 X 
2x3x6 = 15x3x6x4 = 1080. 

Thus, also, any powers of quantities in proportion are in proportion; 
and conversely of the roots. Thus, 

If 2 : 3 : : 6 : 9 then 2* : 3' : : 6'-* : 9' 
2 : 3 : : 6 : 9 „ \/2 : -^3 : : -v/6 : v 9 
2 : 3 : : 6 : 9 „ 2^ : 3' : : 6^ : 9' 

RULE OP THREE. 

When the elements of a problem may be so disposed that they 
form a proportion of which the quantity sought is the last term ; that 
is, when the first bears the same proportion to the second as the third 
does to the fourth or unknown quantity, its value may be easily 
determine<l, and the problem is said to belong to the Goklen EuUj or 
Bute of Three. 

Etdle. — Of the three given terms set down that which is of the 
same kind as the number sought, then consider from the nature of the 
problem whether this number will be greater or less than the term 
so put down ; if greater^ write on its right hand the greater, if leu^ 
the lesser, of the two remaining terms, and place the other on its left 
hand. 

Then multiply the second and third terms together, and divide their 
product by the first; the quotient will be the number sought It 



CEAP. I.] 



BULB OF THR£B. 



85 



moat be observed, that the first and third terms must be reduced to 
tlie nine deDomiiiation ; and if the second term is a compound 
nomber, it should be reduced to the lowest name mentioned ; unless 
the ibird term is a composite number, in which case it is generally 
better to multiply the second term (without any previous reduction) 
h the componeDt parts of the third, as in compound multiplication, 
ifler which divide the compound product by the first term, or by its 
&ctoi8. The answer will be of the same denomination as the second 



tern. 



Examples, 



1. If 3 gallouB of brandy cost 2. How much brandy may be 
19i.,what will 126 gallons cost bought for 39/. 18«., at the rate of 
It the same rate ? { 3 gallons for 1 9 shillings ? 



fA, t. gal. ! 

3:19:: 126 : ? 
19 

1134 
126 

3)2394(798 sbillings 

21 

— or 39/. 18«. Ans. 

29 ' 

27 

24 
24 

3. If 21 yards of cloth cost 
2*^ 10#., what will 1 60 yards cost ? 

frfi. £ «. yds. 

Hefe,21:24 10:: 160 : ? 

4 

4x4x 10=160 

98 
4 



19 



gaL 
: 3 : 



18 



? 



21 



392 
10 

(3)3920 



£ 
39 
20 

798*. 
3 

19) 2394 ( 126 ^wj. 

49 
38 

114 
114 

4. If by selling cloth at 1/. 2«. 
per yard, 10 per cent, is gained, 
what would be gained if it had 
been sold at 1/. 5b. per yard ? 

£ t. £ 9. 

Here, 1 2: 110:: I 5 : ? 
20 20 



22 



25 

no 

2750 



22 



[7)1306 13 4 



£186 13<. 4d. Ana. 



(2) 



111 )1375 



Amount £ 1 25 
Deduct 100 

Gain per cent. £ 25 

= d2 



3^ 



aru -:f thru. 



[PABT I. 



of 5'».'. :-:-r 5 Te^rsw s: 4 

£ ^' £ 
Here. H» : 4 : : o^M 

4 

l<Ni 2244? 


per 


^. If 100 workmen can finish 

a piece of work in 1 2 days, how 

manj men working equally hard 

would hare finbhed it in 3 davs ? 

</. ir. d. 

12: 100:: 3 : ? 

12 

3)1200 

Aftneer 400 workmen. 


£22-4 
2*1 


80 






Interest for I vear, £22 $# 






Then 1 : 22 S : : 5 
5 






;fll2 rt^u^rvr. 





A distinct rule is usually given for the working of problems in 
Compound Frojxtrtion ; but they may generally be solved with 
greater mental facility by means of separate statings. Thus : — 



7. If a person travel 300 miles 
in 10 days of 12 hours each, in 
how many days of 1 6* hours each 
may he travel 600 miles ? 

First, if the days ^ ere of the 
same lenj^h, it would be, bv 
simple proportion, 
m. d, m. 
As 300 : 10 : : 600 : 20 days. 
But these would be days of 12 
hours each, instead of 16, of 
which fewer will be required. 
Hence, again, by simple propor- 
tion, 

h. d. h, d. 
As 12 : 20 :: 16 : 15 
So that the answer is 1 5 days. 



8. If a family of 9 persons 
spend 480/. in 8 months, how 
much will serve a family (living 
upon the same scale) of 24 per- 
sons 16 months? 



I First, as 9 : 480 



P- 
24:o£l280. 



But this would only be the 
expense for 8 months. Hence, 
again. 



m. £ 
As 8 : 1280 



171. £ 
As 8 : 1280 : : 16 : 2560, tbe 
expense of the 24 persons for 1 6 
months. 



yote. — The Rule of Three receives its application in questions of 
Interest^ Discount^ Fellowskipy Bartery &c. 



CHAP. I.] 



BBTXRMINATION OF RATIOS. 



37 



BBTERMINATION OP RATIOS. 

To find the ratio of two numberSy A and B, to each other. 

Rule 1. — DiTide JB hy A, then, unit^ or 1 : the quotient, will be 
tbe iim ratio ; and if the quotient be an integral number, the ratio 
will be expressed in the least terms possible. If, however, it contain 
a fiMOoD, proceed as follows : — first write, 1 : integral portion of 
Ike quUient -f the numerator of the fraction ; then, add 1 to the 
integral, and subtract the numerator of the fraction from the de- 
nominator, and write under the former, 1 : integral portion 4-1 — 
tk 's.ofihe numerator and denominator. 

Then, if the numerator and this difference bo nearly equal, add 
the two ratios together ; but if the numerator and difference are not 
nearly equal, divide the greater by the less ; then, multiply that ratio 
ending with either the numerator or the difference, whichever was 
the divisor, by the integral portion of this last quotient, and add to it 
the other ratio. With the three ratios thus obtained proceed in the 
same manner, from whichever two have the numbers appended by 
the signs -|- and — , nearest equal, to obtain a fourth ratio, and thus 
continue, until this appended number has been eliminated. 

Note. — The ratios thus found will be alternately greater and less 
than the true one, but continually approaching nearer to it. And 
that is the nearest in small numbers, which is immediately followed 
bj much larger numbers : the excess or defect of any one is equal 
to a fraction, having the number appended to that ratio by the sign 
+ or — for its numerator, and the denominator belonging to the 
first quotieDt, for its denominator. 

Example \, 

To find the ratio of 10000 {A) to 7854 {B) in small numbers. 

7854 H- 10000 =0t7^«5^ 



Then, 





Ist 


I. 


2Dd 




3rd 


II. 


4th 


III. 


5tb 


IV. 


6th 


V. 


7th 




8th 


VI. 


8th 


VII. 


10th 


VIII. 


nth 



1 : + 7854 or ratio of 1 to the integer and num. 
1 : 1—2146 or integer 4-1— the '^ of num. and den. 

2146)7854(3 
d : 8—6438 or 2nd ratio x by 3. 
4: 3 + 1416 or 1st and 3rd ratios added together. 
5 : 4— 730 or 2nd „ 4th „ „ 

9 : 7+ 686 or 4th „ 5th „ „ 

14: 11— 44 or 5th „ 6th „ „ 

44)686(15 
210 : 165— 660 or 7th ratio x by 15. 
219 : 172+ 26 or 6th and 8th ratios added together. 
233: 183— 18 or 7th „ 9th „ 

452: 355+ 8 or 9th „ 10th „ „ 

8)18(2 



38 



DETERMINATION OP RATIOS. 



[PABT 



IX. 
X. 



12th 
Idth 

Uth 
15th 



904 : 710-f 16 or llth ratio x by 2. 

1137 : 893— 2 or 1 0th and 1 2th ratios added together. 

2)8(4 

4548 : 3572— 8 or 13th ratio x by 4. 

5000 : 3927 -f or llth and 14th ratios added together. 



The ratios are numbered according to their convergence, with 
Roman numerals to the left hand, and are as follows : — 



± A Ji 

17 3 > T) 



A ±Jl AX9 AAA AAJL 

7> 1I> 112 9 183> 366> 



OJLl 5000 

853 > 3 92 T ' 



Of these the nearest in small numbers is -J^, differing from the 
true ratio by only j-^^Q-Qt and is indicated (as alluded to in the 
preceding note) by being immediately followed by the much larger 
numbers f^^. 

Example 2. 
To find the ratio of 268*8 to 282 in the least numbers. 

2688) 2820 (l-,VftS 
2688 



Then, 



I. 


l8t 

2nd 


II. 


3rd 
4th 


III. 

IV. 


5th 

eth 

7th 


V. 


8th 
9tb 



1 
I 

19 
20 

40 
41 
61 

183 
224 



132 



: 1 -h 132 or ratio of 1 to the integer and num. 
: 2—2556 or integer -|- 1— the '^ of num. and den. 
132)2556(19 

: 19 + 2508 or 1st ratio x by 19. 
: 21 — 48 or 2nd and 3rd ratios added together. 
48)132(2 

: 42— 96 or 4th ratio x by 2. 
: 43+ 36 or 1st and 5 th ratios added together. 
1 2 or 4th „ 6th ,, „ 



: 64— 
12)36(3 
: 192— 
: 235 



36 or 7th ratio x by 3. 

or 6th and 8th ratios added together. 



Therefore the several ratios are -J-, ?% -J^, J|, and |j^. And 
the excess or defect of any one is seen by inspection; thus, -*^ differs 
from the true ratio only -jj^; and -|^, but -^11-^. 

Rule 2. — Divide the greater number by the less, and the divisor 
by the remainder, and the last divisor by the last remainder, and so 
on till remain. Then, 

1 divided by the first quotient, gives the first ratio : 

And the terms of the first ratio multiplied by the second quotient, 
and 1 added to the denominator, give the second ratio : 

And in general the terms of any ratio, multiplied by the next quo- 
tient, and the terms of the foregoing ratio added, give the next suc- 
ceeding ratio. 



CHA?.I.] DBTSIUCINATION OV &ATIOe* 39 

Example 3. 

LetthenoTObcrs be 10000 and 31416, or the ratio Hf?#- 
10000)31416(3 
30000 



1416)10000(7 
9913 



88)1416(16 

88 

536 

528 



Tben, 



8)88(11 

88 



-< s= Ist or least ratio. 



17 7 7 

i Y 7 — and = — = 2nd ratio. 

8 ^ 21 21 ^- 1 22 

7 112 , 112 + 1 113 Q , ,. 

__Y ift— and • — = s=3rd ratio. 

22''^^'" 862 352 4-3 355 

lL^n-'^^ and i?l£±:L = l?^=4tli ratio. 
S65 "^ ^^ " 3905 3905 + 22 3927 

JEfzample 4. 
The mtio of 268*8 to 282 is required. 
2688)2820(1 
2688 

132)2688(20 
264 

48)132(2 
96 

86)48(1 
86 

12)36(3 
36 



40 LOGARITHMIC ARITHMETIC. [PABT 

Then, 

- := 1st ratio. 

- X 20=- and ^^^^ = ^^ = 2nd ratio. 

20 ^ 40 , 40 4- 1 41 « , . 

21 ^ ^=42^"^^ IT^l = 43 =^r<l-^- 

41 , 41 ,41+20 61 , ^ . 

■TT- X 1 = — and 7- — - = -- = 4th ratio. 

43 43 43 + 21 64 

61 „ 183 ^ 183 +41 224 ,^ . 

gj X 3 = — and j^^-^^ = — = 5th ratio. 



Sect. XI. Logarithmic Arithmetic. 

As the nature and properties of logarithms are described in a sub- 
sequent part of this work, being so placed because such description 
could not have been properly understood without a certain acquaint- 
ance with algebra, we shall here only explain the use of the tables 
given in the Appendix, and the method of employing logarithms to 
facilitate the common processes of arithmetic. 

By an inspection of Tabic II., which contains the logarithms of all 
numbers from 1 to 100, it will be seen that each logarithm consists 
of two distinct parts, separated by a decimal point ; thus, the loga- 
rithm of 13 is 1*113943; the number to the left of the decimal 
point (or 1 in the above example), is called the index or charac- 
teristic*^ and its value depends only upon the number of digits in the 
quantity whose logarithm it is, without any regard to thet7a/tt€ of that 
quantity, and it is always 1 less than that number of digits ; thus, in 
the example, the characteristic of the logarithm of 13, which contains 
two digits, is 1, or one less than that number ; and it will be seen from 
the Table, that 1 is the characteristic of all the logarithms from 10 
to 99, but that, for numbers below 10, the index is 0, and for 100 
is 2, in each case 1 less than the number of digits in the quantity of 
whose logarithm it is the characteristic. The characteristic, there- 
fore, of the logarithms of all numbers 

equal to or greater than 1 and less than 10 is 0* 



10 


» 


100 „ 1- 


100 


>» 


1000 „ 2- 


1000 


>» 


10000 „ 3- 


10000 


»> 


100000 „ 4- 


&c. 




&c. &c. 



* In order to avoid confusion from the use of the leord index to signify two 
thincB, we shall throughout this work employ the term characterisHc when 
speaking of logarithms, and iruies when speaking of roots or powers. 



CHAP. I.] LOGARITHMIC ARITHMETIC. 41 

When the qaantity is less than unity, the characteristic of its loga- 
rithm becomes negative, and its value is determined hy the number 
of cyphers which occur between the decimal point and the first signi- 
ficant figure, (the fraction being decimally expressed), and is always 
1 greater than such n amber of cyphers; or it is equal to the differ- 
ence in the number of figures in the numerator and decimal denomi- 
nator; thas, the characteristic of the logarithm of 



•1 


or 


iV 


is 


1-^ 


•01 


» 


lio 


»» 


2- 


•001 


» 


16^6 


>» 


3- 


0001 
&c. 


>» 


1 


?» 


4. 


10000 

&c. 


&c. 



The decimal part of the logarithm, or that lying to the right of 
the decimal point, depends entirely on the relative value of the figures 
composing the quantity whose logarithm it is, and not at all upon the 
tttoal numerical value of that quantity ; thus, in the example already 
giTen, the decimal part of the logarithm of 13 is '113943, which is 
iko Ae decimal part of the logarithm of 1*3, or 130, or 1300, for in 
each case the 1 and the 3 have the same relative value. So that the 
decimal portion of a logarithm is always the same for the same 
fignrea, and is not altered by the addition of any number of cyphers 
either to the right or to the left hand of those figures, or what is 
equivalent, by the multiplication or division of the quantity by 10, 
or any power of 10 ; it is only the characteristic of the logarithm 
which alters its value, 1 being akded to the characteristic for every 10 
hy which the quantity is mvltipliedy or subtracted from it for every 10 
hy which the quantity is (/«t7M/e(/. Thus, 



the logarithm of 745800 


being 5872622 


that of 74580 


is 


4-872622 


„ 7458 


>» 


3-872G22 


745-8 


>> 


2-872622 


74-58 


»> 


1-872622 


7-458 


» 


0*872622 


•7458 


99 


1-872622 


„ 07458 


>» 


2-872622 


•00745fi 


^ » 


3-872622 



It must be borne in mind, that in the logarithm of a fractional 
qnantity, it is only the characteristic which has a negative value, and 
that the decimal pal^t of a logarithm is always positive. It is, how- 
CTer, sometimes convenient to have the whole logarithm expressed 
negatively, both characteristic and decimal ; for which purpose, sub- 

• The negative sign ( — ) ia always placed above the characteriatic, thus 2, 
initcad of before it, in order to avoid its being misunderstood for the sign of 
mbcractiim. 



42 LOGARITHMIC ARITHMETIC. [PART I. 

tract the last right hand figure in the decimal portion from 10, and 
all the others from 9, and the result will he what is termed the 
arithmetical complement of the decimal, to which prefix the former 
characteristic less I, and the result will he a negative logarithm, 
equivalent in value to the original logarithm having only a negative 
characteristic; for example, the logarithm of '07458, as above, is 
2^-872622, which is equivalent to — 1 127378. It is also frequently 
convenient to take the arithmetical complement of the whole 
logarithm, and this is obtained by subtracting the right hand figure 
of the decimal from 10, and all the others from 9, including the 
characteristic when positive^ but if negative it must be added to 9. 
Thus, the arithmetical complement 

of 3-14G128 is 6-853872 
„ 207G276 „ 11-923714 
„ 5-322839 „ 4-677161 
„ i-986772 „ 10013228 

USB OP THE TABLES. 

To find the logarithm of any given number. 

If the number is less than 100, its logarithm will be found in 
Table II., with its proper characteristic prefixed ; but if the number 
contains more than two figures, its logarithm may be found from 
Table III, as follows: — If there are only three figures in the num- 
ber, look for that number in the first column of the table, and on the 
same line in the next column to the right, under 0, will be found the 
decimal portion of the required logarithm, to which the proper cha- 
racteristic must be prefixed, according to the rules which we have 
just explained. If the quantity contains four figures, look for the 
first three figures in the first column as before, and the four last 
figures of the logarithm of the required number will be found on the 
same line with those three figures, and in that column which has at 
its head the fourth figure of the given number ; the two first figures 
of the logarithm will be found in the second column (headed 0), and 
which figures being common to all the logarithms enclosed by each 
pair of horizontal lines, it is unnecessary to repeat. Where these 
first figures change their value in the middle of a line, the same 
is indicated by a break in the horizontal line, thus, 139879 | 0194, 
which shows that the two first figures (13) have changed to 14, and 
the right hand logarithm is therefore 140194. The heading figures are 
repeated at every tenth line in the body of the tables, in order to 
facilitate their use. 

Examples. 

Required the logarithm of 734. 

In Table III., on the same line with 734 and under 0,. are found 
5696, the four last figures of the logarithm^ to which the common 



CHIP. I.] LOOARtTHMIC ABITBlfBTIC. 48 

figures 86 and the proper characteristic 2 being prefixed, we obtain 
2'^569C, the logarithm required. 
Find the logarithm of 3476. 

Hpre, on the same line with 347 and under 6, will be found 1080, 
which, with the two first figures and the characteristic prefixed, is 
3'54108O, the logarithm required. 

The log. of 5-84 is ^766413 

„ 0932 „ 2-969416 

10-24 „ 1 010300 

„ 3708 „ 3-569140. 

When the quantity whose logarithm is required contains more than 
four figures, proceed as follows: — Find the logarithm for the first four 
figures as above, then look in the first column of Table I. for the first 
figures, and on the same line in the column having at its head the 
fifth figure will be found the quantity which must bo added to the 
logarithm already taken out^ to give the logarithm of the quantity first 
required. If the first four figures are not found in the first column 
of the table, then take the line containing the next less number to it. 
If the number whose logarithm is required contains more than ^yc 
figures, proceed as above to obtain the logarithm of the first five 
figures, then, on the same line of Table I. that the number added to 
the logarithm for the fifth figure was found, and in the column 
htTJDg at its head the sixth figure, will be found a quantity, which, 
dirided by 10 (or what is the same, having its right hand fisure taken 
twtj*), and added to the logarithm already found, will give the 
iogtfithm of the first six figures; again, on the same line and in the 
cokmn baring at its head the seventh figure, will be (bund a quan- 
tit?, which, divided by 100 (or baring two figures cut off from the 
right band), and added, will give the logarithm for seven figures t. 

Examples. 

Required the logarithms of 11488, 621547, 768654, 7642179. 

log. of the first four figures from Tab. Ill = 4059942 

From Tab. I. on line with 1148 and under 8 ... = 302 



Logarithm of 11488, as required = 4060244 

Log.of6215 =1= 5-793441 

From Tab. I. on line with 6160, the next less) _ „« 

No. in the tab. to 62)5 and under 4 J "" 

On aame line under 7 = 4 



Logarithm of 621547 = 5793474 

* If the figure thus cot off ezoeeds five, one must be added to the first right 
kind fijnire left. 
t See remark at page 45, with regard to the number of places to be de- 



44 LOOABITHMIC ARITHMETIC. [PART I. 

Log. of 7686 = 5-885700 

From Tab. I. on same line with 7686 under 5... = 28 

On same line under 4 := 2 



Logarithm of 768654 = 58 85730 

Log. of 7642 = 3-883207 

From Tab. I. on same line with 7552 under 1 ... = 5 

On same line under 7 = 3 

On same line under 9 = 



7 

99 

513 



Logarithm of 76421 79 = 3883217 



To find the number answering to any given logarithm. 

Look in Table III. for the given logarithm, or the next less in 
value to it that can be found, then on the same line, in the first 
column, will be found the first three figures, and at the head of the 
column in which the logarithm was found, the fourth figure of the 
number sought. If the given logarithm is found exactly in the table, 
the figures thus obtained will be the required number, care being 
taken to point off one more figure to the left hand than there are 
units in the characteristic of the given logarithm, cyphers being at- 
tached to the right hand of the number, if requisite. If, however, 
the given logarithm is not found exactly in the table, subtract from it 
the next less logarithm found, calling the remainder the first differ- 
ence ; then look in Table I. on the same line with the four figures 
already obtained from Table III. (or the next less figures which can 
be found) for this difference, and at the head of the column in which 
it is found will be the fifth figure of the number sought. If the first 
difference is not found exactly in the table, look for the next less 
number to it, which subtract from the first difference for the second 
difference ; then add a cypher to this second difference, and look for 
it on the same line of Table I. as before, and the figure at the head 
of the column containing the nearest number to it, either greater or 
less, will be the sixth figure of the number required. 

Examples. 
Required the number answering to the logarithm 3*241756. 

Given log. = 3'241 756 
Next less log. in Tab. III. =3*241546 = the log. of 1744* 

210 first dif. 
In Tab. I. on same line with ) i nn • r j • i o o 

the next less No. to 1 7*4 2^ '^ *^"""^ '" *="'• * ]^ 

110 second dif. 
On the same line 99 is found in col. 4 *04 



The No. required = 1744*84 



CHAP. 1.] LOGARITHMIC ARITHMETIC. 45 

In this example the next less logarithm which can he found in 
Table III. is 3*241546, the numher answering to which, 1744, is the 
first four figures of the namher sought ; then subtracting this loga- 
rithm from the given logarithm, we obtain for the first difference 
210, and looking in Table I. on a line with 1740 (the next less 
number to 1744), for the next less number to 210, we find 199, at 
tbe head of the column containing which is 8, the fifth figure re- 
quired; then subtracting 199 from 210, we obtain the second differ- 
ence, 11, and adding a cjrpher, the nearest number which we find on 
the same line is 99, at the head of the column containing which is 4, 
the sixth figure required. 

Required the numbers answering to the following logarithms : — 
3510009, 2-475771, 5-871624. 

The number answering to the logarithm 3*510009 is found at 
oQcetobe32d6. 

Given log. = 2-476771 
Next less log. = 2-475671 = the log. of 299-0 



From Tab. I.... 


100 = 1st dif. 
87 


•06 




130 = 2nd dif. 
130 


-009 




No. required = 

■ 

= 5-871624 

= 5-871573 = the log. of 


299069 


Given log. 
Next less log. 


744000' 


Prom Tab. I.... 


51 = 1st dif. 
46 


80- 




60 
62 


9- 



The No. required = 744089- 

It should be observed here, that the number of figures which may 
^ depended upon in any result obtained by logarithms, will be equal 
Jo the number of decimal places in the logarithms employed ; thus, 
in Qiing the tables appended to this work, the results obtained will 
^ accurate to six figures, except towards the end of the tables, in 
which only five figures should be trusted. 

MULTIPLICATION AND DIVISION BY LOGARITHMS. 

To mulHpiy two numbers together, add together their logarithms, 



4i6 LOOABITHIfIC ARITHMETIC. [PART I. 

and the sum will be the logarithm of their product; or, to divide 
one number by another, subtract the logarithm of the divisor from 
the logarithm of the dividend, and the remainder will be the log- 
arithm of the quotient of the two numbers. 

Ex, — Multiply 80 X 43 X 72*64; and divide the product of 
(7143 X 6278) by 3145. 



Add 



Log. 80 = 1-47712J; Log. 7143 
log. 43 = 1-633468 add log. 6278 



Jog. 72-54 = 1-860578 
Log. 93576-4 = 4-971167 



sub. log. 314-5 



: 3'85S881 
3-797821 

7-651702 
2-497621 



Log. 142589 ;= 5154081 



rROPORTION OR THE RULE OP THRBB BY LOGARITHMS. 

The Rule of Three is very readily performed with the aid of log- 
arithms, by simply adding together the logarithms of the 2nd and 
3rd terms, and subtracting the logarithm of the 1st, the remainder 
being the logarithm of the 4th term, or number required to be 
found. Or, instead of subtracting the logarithm of the 1st term, we 
may €idd its complement, (the method of obtaining which has been 
already explained at page 42,) and subtract 10 from the character- 
istic of the result, which will, as before, be the logarithm of the 
4th term. 

Examples. 

The following are the same as the 1st, 3rd, 4th, and 5th Ex- 
amples given in the Rule of Three, at pages 35 and 36. 



1st Ex. 

Log. 19 = 1-278754 
+ log. 126 = 2-100370 

3-3791-24 
- log. 3 = 0-4771 21 



3rd Ex. 

Log. 245 = 1-389166 
4- log. 160 = 2-204120 



3-593286 
- log. 21 = 1-322219 



Log. 798 =s 2-902003 Log. 186-667 = 2-271067 

4th Ex. 5th Ex. 

Comp. of log. 22 = 8-657577 Comp. of log. 100 = 8000000 

-f- log. 110 = 2041393 + log. 4 = 0602060 

+ log. 25 = 1-397940 + log. 560 = 2748188 



Log. 125 = 2096910 



Log. 22-4 = 1-350248 



CHIP. I.J LOGABITHMIC ARITHlfKTIC. 47 

EVOLUTION AND INVOLUTION BY LOOABITHM8. 

To perfonn the operation of involution^ or the raising of powers, 
it is only necessary to multiply the number, any power of which is 
required, by the index of that power, and the product will be the 
logirithm of the required power; and, inversely, the operation of 
miuiioH^ or the extraction of roots, is performed by simply dividing 
the logarithm of the number by the index of the root required, the 
quotient ynl\ be the logarithm of the root 

Examples. 

Square 84, cube 13, and raise 7 to the sixth power. 

Log. 84 = 1-924279 x 2 = 3-848558 = 7056 = 84- 
log. 13 = 11 13943 X 3 = 3341829 = 2197 = 13» 
log. 7 = 0-846098 x 6 = 5070588 = 117649 = 7^ 

Extract the square root of 576, the cube root of 4913, and the 
axth root of 46656. 

Log. 576 = 2-760422 -r 2 = 1 3802 11 = 24 = n/ 576 
log. 4913 = 3-691347 -^ 3 = 1-230449 = 17 = 'V 4913 
log. 46656 = 4-668908 -r 6 = 0779151 = 6 = V46666 • 

It is necessary here to make a few remarks on performing the 
operations of evolution and involution on logarithms with negative 
cWacteristics. In doing this, it must be borne in mind that it is 
only the characteristic which has a negative value, the decimal part of 
tie logarithm being always positive ; therefore, if it is required to 
multiply a logarithm with a negative characteristic by any number, 
iim multiply the decimal part of the logarithm, pointing oflf as many 
decimal figures in the product as there were in the logarithm, then 
multiply the characteristic, and subtract from the product the num- 
ber (if any) pointed off to the left in the first product, the result will 
W the negative characteristic, and the decimals pointed off in the 
first product will be the decimal part of the required logarithm. 



of 



Examplei. 
Required the square of *25, the cube of '375, and the sixth power 

The logarithm of -25 = T-397904 
First multiply 897940 by 2 
2 



_ _-795880 

Then 1x2=2 



2-795880 = -0625 = -25*. 



48 LOGARITHMIC ARITHMETIC. [PART I. 

The logarithm of -375 is f-574031. 
•574031 
3 



1-722093 
1 X 3 = 3- 



2-722093 = -05273437 = •375». 



The logarithm of 7 is 1-845098 
-845098 
6 



5-070588 
1x6 = 6- 



1-070588 =-117649 = •7«. 



To divide a logarithm with a negative characteristic by any nam- 
ber :— If the characteristic is a multiple of that number, that is, if k 
is divisible by it without remainder, proceed as in ordinary division ; 
if not, separate the characteristic from the decimal, and add to the 
characteritttic a number which will make it divisible^ and prefix to the 
decimal the same number, then divide both by the given divisor, 
and the quotients will be the characteristic and decimal of the 
logarithm required. 

Examples, 

Required the square root of -0625, the cube root of -74, and the 
fifth root of -543. 

The logarithm of -06*25 is 2-795880 
Then 2)2-795880 



r397940 = -26 = \/0625 

The logarithm of -74 =1-869232. 

Then 1+2=3-7-3 = 1- 

and 2-860232 H- 3 = -956411 



V-74 = -9045 = 1-956411 

The logarithm of -543 is 1-734800 

Then T -f 4=5-r-5 =1- 

and 4-734800 -f- 6 = -946960 



V-543 = -885034 = 1946960 



CRIP. 1.] PBOPRRTIRS OF NUMBERS. 49 

Sbct. XII. Properties of Number m. 

To render these intelligible to the student, we shall here collect a 
few definitions. 

Def. I. A tintV, or unity ^ is the representation of any thing con- 
adered individoally, without regard to the parts of which it is com- 



2. An integer is either a unit or an assemblage of nnits ; and a 
Jhetum is any part or parts of a unit. 

3. A multiple of any number is that which contains it some exact 
namber of times. 

4. One number is said to measure another, when it divides it with- 
out leaving any remainder. 

5. And if a number exactly divides two, or more numbers, it is 
then called their common measure, 

6. An even number, is that which can be halved, or divided into 
tvo equal parts. 

7. An odd number, is that which cannot be halved, or which differs 
from an even number by unity. 

8. A prime number, is that which can only be measured by 1 , or 
onitT. 

9. One number is said to be prime to another when unity is the 
odv Dnmber by which they can both be measured. 

10. A composite number, is that which can be measured by some 
Dumber greater than unity. 

11. A perfect number, is that which is equal to the sum of all its 
difigore, or diquot parts : — thus 6= f + f + f • 

Pnp. 1. The sum or difference of any two even numbers is an 
tten number. 

2. The sum or difference of any two odd numbers is even ; but the 
nm of three odd numbers is odd, 

3. The sum of any eifen number of odd numbers is even ; but the 
nm of any odd number of odd numbers is odd, 

4. The sum or difference of an even and an odd number is odd, 

5. The product of any number of even numbers is even ; and any 
power of an even number is even, 

6. The product of any number of odd numbers is odd; and every 
power of an odd number is odd, 

7. The product of any number of even numbers, by any number of 
<^nombers, is even. 

8. An odd number cannot be divided by an even number, without 
t remainder. 

9. If an o</<ar number divides an even number, it will also divide the 
Wfofit 

10. If a number consist of many parts, and each of those parts 
^n a common divisor d, then will the whole number, taken col- 
^▼ely, be divisible by d, 

11. Neither the sum nor the difference of two fractions, which are 
n their lowest terms, and of which the denominator of the one con- 

E 



50 PR0PBRT1B8 OF NUUBRR8. [PART T. 

tains a factor not common to the other, can be equal to an integral 
number. 

12. If a square number be either multiplied or diTided by a square, 
the product or quotient is a square ; and conversely, if a square num- 
ber be either multiplied or divided by a number that is not a square, 
the product or quotient is not a square. 

1 3. The product arising from two different prime numbers cannot 
be a square number. 

] 4. The product of no two different numbers prime to each other 
can make a square, unless each of those numbers be a square. 

15. The square root of an integral number, that is not a complete 
square, can neither be expressed by an integer nor by any rational 
fraction. 

IG. The cube root of an integer that is not a complete cube cannot 
be expressed by either an integer or a rational fraction. 

17. Every prime number greater than 2, is of one of the forms 
4n 4- 1, or 4» — 1. 

18. Every prime number greater than 3, is of one of the forms 
6n -r 1, or G w — 1. 

1 9. No algebraical formula can contain prime numbers only. 

20. The number of prime numbers is infinite. 

21. The first twentv prime numbers ore 1, 2, 3, 5, 7, 11, 13, 17, 
19, 23, 29, 31, 37, 4f, 43, 47, 53, 59, 61, and 67. 

22. A square number cannot terminate with an odd number of 
cyphers. 

23. If a square number terminate with a 4, the last figure but one 
(towards the right hand) will be an even number. 

24. If a square number terminate with 5, it will terminate with 25. 

25. If a square number terminate with on odd digit, the last figure 
but one will be eren ; and if it terminate with any even digit, except 
4, the last figure but one will be odd, 

2G. No square number can terminate with two equal digits, ex- 
cept two cyphers or twoybwr*. 

27. No number whose last, or right-hand digit is 2, 3, 7, or 8, is 
a square number. 

28. If a cube number be divisible by 7, it is also divisible by the 
cube of 7. 

29. The difference between any integral cube and its root is always 
divisible by 6. 

30. Neither the sum nor the difference of two cubes can be a 
cube. 

31. A cube number may end with any of the natural numbers 
1, 2, 3, 4, 6, 6, 7, 8, 9 or 0. 

32. If any series of numbers, beginning from 1, be in continued 
geometrical proportion, the 3rd, 5th, 7th, &c. will be squares ; the 
4th, 7th, 1 0th, &c. cubes ; and the 7th, of course, both a square and 
a cube. 

33. All the powers of any number that end with either 5 or 6, 
will end with 6 or 6, respectively. 



CHAP. 1.] PROPERTIES OF NUMBRRS. M 

U. Any power, w, of the natural numbers, 1, '2, 3, 4, 5, 0, &c. 
las as many orders of differences as there are units in the common 
exponent of all the numbers; and tlie last of those differences 
U a constant quantity, and equal to the continual product 

Ix*2x3x4x xw, continued till the last factor, 

or the number of factors be w, the exponent of the powers. Thus, 

Tbe 1st powers I, Si, 3, 4, 5, &c., have but one order of 
differences 1111 &c., and that difference is 1 . 

The iind powers I, 4, 9, 16, 25, &c., have two orders of 
(Ufferences 8 5 7 9 

of which tbe last is constantly 2 = 1 x 2. 

The 3rd powers 1, 8, 27, 04, 125, &c., have three orders of 
differences 7 19 37 61 
12 18 24 
6 6 
of which the last is 6 = 1 X 2 X 3. 

In like manner, the 4th, or last, differences of the 4th powers, 
•re each = 24 = 1 x 2 x 3 x 4 ; and the 5tb, or last differences 
of the 5th powers, are each 125 =: 1 x 2 x 3 x 4 x 5. 

35. If unity be divided into any two unequal parts, the sum of 
the square of either of those parts added to the other is the same. 
Thosj of tbe two parts ^ and ♦, | + (f )« = ^ -h (\)- =: ^ ; so, 
•g«n, of the parts J and ^, ?- + (f)« = -J + {fY = if * 

For tbe demonstrations of these and a variety of other properties 
of nmnbers, those who wish to pursue this curious line of inquiry 
Bttjconsalt Legendre "Sur la Theorie des Nombres," the "Dis- 
<iaifltioDet Arithmeticse" of Gauss, or Barlow's " Elementary Inves- 
tigition of the Theory of Numbers." 

Alto, for the highly interesting properties of Circulating Decimals^ 
ttd their connexion with prime numbers^ consult the curious works 
rf the late Mr. H. Goodwyn, entitled " A First Centenary," and " A 
Ttble of the Circles arising from the Division of a Unit by all the 
htesenfrom 1 to 1024." 



K 2 



52 DEFINITIONS [pART I. 



CHAP. II. 

ALGRBRA. 

Skct. I. Dejinitunis and Notation. 

Algebra is the science of the computation of magnitudes in gene- 
ral, as arithmetic is the particular science of the^ computation of 
numhers. 

Every figure or arithmetical character has a determinate and indi- 
vidual value ; the figure 5, for example, represents always one and the 
same number, namely, the collection of 5 units, of an order depend- 
ing upon the position and use of the figure itself. Algebraical cha- 
racters, on the contrary, must be, in general, independent of all par- 
ticular signification, and proper to represent all sorts of nambers or 
quantities, according to the nature of the questions to which we apply 
them. They should, moreover, be simple and easy to trace, so as to 
fatigue neither the attention nor the memory. These advantages are 
obtained by employing the letters of the alphabet, a, A, c, &c. to 
represent any kinds of magnitudes which become the subjects of 
mathematical research. The consequence is, that when we have 
resolved by a single algebraical computation all the problems of the 
same kind proposed, in the utmost generality of which they are sus- 
ceptible ; the application of the investigation to all particular cases 
requires no more than arithmetical operations. 

It is usual, though by no means absolutely necessary, to represent 
quantities that are known by the commencing letters of the alphabet, 
as a, ^, c, dy &c., and those that are unknottn by the concluding letters 
w, Xy t/y z. But it is often convenient, especially as it assists the 
memory, to represent any quantity which enters an investigation, 
whether known or unknown, by its initial letter ; as Bum by «, pro- 
dtict by py density by rf, velocity by c, time by t; and so of others. 

In addition to the signs already explained as being used in arith- 
metic, the following symbols and modes of expressing certain opera- 
tions, are employed in algebra. 

The product of two or more quantities is expressed either by in- 
terposing the sign of multiplication, as a x b x c x d; or by inter- 
posing dotSy which have the same signification, bs a . b . c .d; or, more 
simply, by placing the letters merely in juxtaposition, B&abcd. And 
as it is immaterial in what order the multiplication is performed, it is 
usual to write the letters after each other in the order in which they would 
stand in the alphabet, placing any figures which may occur in the product. 



THiP. 11.] AND NOTATION. 53 

at the commencement. Thus, the continual product of /2 ^, a;, 7 e, 
lod S c, may be written either ^bzl eScy or x ^bl eSc^ &c., but 
it is preferable and usual to write them i^ b e e x. The figures 
% 3, 7 and 42, by which the letters are multiplied, are termed their 
mficienti; when a letter is not preceded by any figure (as x in the 
example), its coefficient is 1, or unity. 

We give the name term to any quantity separated from another by 
the sign -I- or — . When an algebraical expression consists of only 

1 term, as 4 a it is called a monomial. 

2 „ ac — 4 a & . . . „ binomial*. 
8„ o + fc — a c . , ' „ trinomial. 

4 „ c + </— Sy-H X . „ quadrinomial. 



{multinomial^ 
or polynomial . 



The signs + and — , which in arithmetic simply indicate the opera- 
tioos of addition and subtraction, are employed more extensiyely in 
algebra, to denote, besides addition and subtraction, any two opera- 
tioiis or any two states which are as opposed in their nature as addi- 
tkn tnd subtraction are. And if, in an algebraical process, the 
i^ + is prefixed to a quantity to mark that it exists in a certain 
itete, pontion, direction, &c., then, whenever the sign — occurs in 
eoDoexion with such quantity, it must indicate precisely the con- 
tiary state, position, &c. and no intermediate one. This is a matter 
of pare convention, and not of metaphysical reasoning. Other cha- 
rKters might have been contrived to denote this opposition ; but they 
would be superfluous, because the characters + and — , though ori- 
gioally restricted to denote addition and subtraction, may safely be 
otended to other purposes. 



■ignifies any^ ^ theright, \ nifies \ to the left, 
(, forwards, J (.backwards. 



|. . / Increase, "^ — a signi- ( Decrease, 

^^^ ^ J Money due, f respond- j Money owing, 

^"^^°*^^ ( Motion upwards, ) mg v Motion doT^nward. 

And so on in every species of contrariety. And two such equal quan- 
tities connected togeUier in any case destroy each other's effect, or are 
«|8al to nothing, as + a » a'= 0. Thus, if a man has but 10/. 
ad at the same time owes 10/. he is worth nothing. And, if a ves- 
•d which would, otherwise, sail six miles an hour, be carried back 
a miles an hoar by a current, it makes no advance. 

* When the second term of a binomial has the negative sign ( — ) it is called 



54 ADDITION AND SUBTRACTION. [PART T. 

Like Bigns are either all poMtive (-f ), or all negative ( — ). Aud 
unlike are when some are positive and others negative. If there be 
no sign before a quantity, the sign + is understood. 

Like quantities are such as contain the same letters, and differ only 
in their coefficients^ asa — 7 a, or9a6j:+ 2abx^^abx, Un- 
like quantities are such as contain different letters, as a 6 — c/ + bkx. 

When any number of terms are collected under a vinculum or bar, 
thus, a -f /» — c/^, or inclosed in parentheses thus, (a + 6 — c/)*, 
it denotes that the whole quantity so enclosed is to be taken col- 
lectively, and subjected to whatever operation or process is indicated 
by the symbol without the bar or parenthesis ; thus, in the example, 
that the compound quantity a -f 6 — cfy is to be squared. 

An equation is when two sets of quantities which make an equal 
aggregate are placed with the sign of equality ( = ) between them ; 

As 12 4- 5 = -40 — 3, or X -f ;/ = a + ft — c d. 

The quantities placed on both sides the sign of equality are called 
respectively the inemhcrs of the equation. 

The sign x indicates that the quantity before which it is placed 
is infnite^ or unlimited in its value. 

The symbol a , placed between two quantities, signifies that the 
second varies as the first : thus, a 7 /> is read a varies as b. 

The word therefore being of frequent occurrence in deducing the 
successive steps of algebraic processes, is denoted by . • . . 



Sect. II. Additimi and Subtraction, 

In algebra, the operations answering to those of addition and sub- 
traction in arithmetic cannot with propriety be called by those names, 
as either of them in algebra frequently involves the actual use of both 
those processes. They would be better expressed by some general 
term denoting the incorporation or striking a balance between the 
several quantities employed. 

When two simple quantities, or monomialsy are to be added to- 
gether, as 2 ft and c, or 4 a and 5 a, it is done by connecting them 
together with the sign of addition (-f); as2ft + c, 4a-|-6a; but 
when the quantities are similar, as in the second example, the expres- 
sion may be simplified by adding together the two coefficients, and 
subjoining the common letter, which would then become 9 a. 

In like manner, the subtraction of simple quantities, or monomialsy 
may be expressed by the sign of subtraction ( — ), or in the case of 
simple like quantities by actually subtracting tlie coefficients, and 
subjoining the common letter ; thus, the subtraction of ft from a, 
/from c, 4 ft c from 7 ft c, is performed thus, a — ft, c — /J 3 ft c. It 
not unfrequently occurs that the quantity to be subtracted is greater 
ihan the quantity from which it is to be taken, the consideration of 
which being totally different from all that the learner has been accus- 
tomed to in arithmetical operations, involves a difficulty in its appro- 



CBAP. IL] addition AND SUBTRACTION. 55 

bension, bat this will soon be remoTed by an attentive perusal of the 
piragraph at page 53, explaining the sense in which the signs 
+ and — are used in algebra. Where such occurs, the less quan- 
tity must be taken from Uie greater, and the negative sign prefixed to 
the remamder. 

To add together compound quantities, or polynomials, as 
(o + 2 6-1- c) and (4 o -f 5 c — rf), write one after the other, 
with their proper signs attached, omitting the parentheses, as 
fl + 2ft-fc + 4a 4- 5c— </, which may be simplified by the fur- 
ther addition or subtraction of its separate terms, and then becomes 
5a + 26+ 6 c — </. 

Wben quantities to be added are presented promiscuously, it is best 
to classify them preyious to their addition. 

Thus, 3 aS — 3 * c, + 2 c2, + 4 ^, 3 a^ -h 5 6c - 2 c^ 

+ 7aS +56c, -f aS — 2cS —46c, 7 a« —86c -f 4rf 

when arranged become as in the margin, a^ _ 4 6 c + 2 c^ 
and their sum is readily obtained, as in 



thefoorth line. 11 a« — 2 6c + 4 rf 



To subtract one compound quantity from another, change the sign 
of each term of the subtrahend^ and then proceed as in addition. 

Thus, 4 « 6 — 3 6 c And 4a6 — 3c«-f 6c 

— (2 a 6 — 6 6 c) «. ( a 6 — c^ — 2 6 c) 



become 4 a 6 — 3 6c become 4a6 — 3c*+ 6c 

— 2a6-f66c — tf6-f c'^+26c 



Resalt 2 a 6 -h 3 6 c Result 3a6 — 2c« + d6c 



The reason of changing the signs of the subtrahend may be ex- 
plained as follows : — Let it be required to subtract (c— cQ from (a -f 6) ; 
let us first subtract c, and the result will be a + 6 — c ; but it is evi- 
dent that in subtracting c from (a -f 6), we have taken away more 
than we ought, since it is only c— d which should have been sub- 
tncted, we have therefore taken away too much by </, and to obtain 
the correct result, must add it again to a + 6 — c, which then bc- 
eomes a + 6 — c + </> in which it is obvious that the signs of the 
nbtrahend (c — d) are changed. 

In addition and subtraction of algebraic fracti(m8^ the quantities 
Buist be reduced to a common denominator, and occasionally undergo 
other reductions similar to those in vulgar fractions in arithmetic; 
>Qd thee the sum or the difference of the numerators may be placed 
om the common denominator, as required. 



56 MULTIPLICATION. [PART I. 

m, « . c ad be ad + be 

..a b e a b a e be 

And, J-- 4.— +— ^-1- — 4.4.-.= 
o c ae a b c b a 

a2 b^ <r a""' b^^ a« c« b^c" 

aoc abc abc abe abc abe 

a« + *• + c« + a' ft" 4- a'^ c- + ft^ ^^ 
a ft c 
. , a c ad ^ bc^ 

a — iT ac + c^ arf — dx 



c cd c d 

ac -j- ex -^ ad + dx _ a{C'-d) -f ^ (c + rf) 
ed ~ ed 

And, *+* *-* 



ft — X ft 4- 0? 
(ft2 4- aft« + ^«) - (ft« - 2fta? 4- ««) 4 ft. 



ft- - a?2 "■ ii ^0.2' 



Sect. III. Multiplication. 

1 . To multiply one monomial by another, multiply their coefficients 
for the coefficient of the product, and subjoin to it the letters com- 
posing the two quantities in their natural order, and if the same let- 
ter occurs in both, add together the two exponents for its exponent 
in the product, 1 being understood where no other exponent is ex- 
pressed. In attaching the proper sign to the product, observe that 
the product of two factors having contrary signs is negative^ and of 
two factors having the »ame sign positive. Or briefly, that like signs 
give 4-9 and unlike signs — . 

Note. — The general rule for the signs may be rendered evident 
from the following definition ; multiplication is the finding a magni- 
tude which has to the multiplicand the proportion of the multiplier 
to unity. Hence, the multiplier must be an abstract number, and, if 
a simple term, can have neither 4- nor — prefixed to its notation. 
Now first, 4- a X 4- »» = 4- »» «> for the quality of a cannot be 
altered by increasing or diminishing its value in any proportion; 
therefore the product is of the quality pltUy and m a by the definition 
is the product of a and m. Secondly, — ax 4-m = — mo, for 
the same reasons as before, mutatis mutandis. Thirdly, 4- a x — m 
has no meaning ; for m must be an abstract number, therefore here 



CHIP. II.] MULTIPLICATION. 57 

we can baTe no proof. Bat + ax (m -^ n) s= ma ^ na^ n being 
Jess than m; for a taken as often as there are units in m is = m a 
by the first case ; bat a was to have been taken only as often as there 
ire units in m — n; therefore a has been taken too often by the units 
in n; consequently a taken n times, or n a, must be subtracted ; 
sod of course ma — n a is the true product. Fourthly, 
— ax (m — n)a=: — ma -f na. For -^ a x m s= — ma (by 
case 2); but this, as above, is too great by — na; therefore — ma 
with n a subtracted from it is the true product ; but this, by the rule 
of subtraction, is = — wi a + w a. 

2. To find the product of two pdynomiahy multiply each term of 
the one mto all those of the other, following the rule given for mo- 
uomials. 

3. To multiply algebraic fradionSy take the product of the nume- 
rators for the new numerator, and that of the denominators for the 
new denominator. 

Examples. 

1. iah X bed =i ^ . 5 . ah .cd z= ^0 abed, 

2. 8 a« ^» X 4 a* 6 = 8 . 4 . a2 . a* . 6' . * = 32 a« + * 
fc' + i = 32o» bK 

3. Multiply 2o + *c-2*« 

By2a — *c + 2ft« 



4 a« + 2 a * c — 4 a 5« 

— 2a*c— 6«c« +2 5«c 

+ Aab^ +2 5»c — 4 6* 

Product, 4 a« — *« c« +4^«c--4 i^* 



U + fc l» -H 



2 a 6 + ft* 
b 



o«+ ab a» +2a« b + ab* 

ab -{- b* + a«i^ + 2ad« + ft* 



a« + 3a6 + ft« a» •f3a«ft + 3aft« +6» 



a + ft 
ft 



••{:i 



« +aft 
- a ft - ft« 

a« -ft« 



58 DIVISION. [part I. 

a + b a ^b _ ( g -j- 6) (a — b) _ a" — b\ 
c rf"" c X d ^ c d 

2« "dab 3ac__ IS a- be x __ 9 a j? 
*o c ^b " tiabe "l 

- — — - = 9aa?. 

iVb/tf. — From the above examples (4, 5, and 6) we may Icani — 

1. Tliat the square of the sum of two quantities is equal to tbe sum 
of tbe squares of tbe two quantities together witb twice their pro- 
duct. 

2. That tbe cube of tbe binomial a -^ by is a* -f S a' b + 
'dal^ -{• b\ 

3. That the product of tbe sum and difference of two quantities is 
equal to tbe difference of their squares. 



Sect. IV. Division, 

1. To divide one monomial by another, divide the coefficient of the 
dividend by tbe cocfRcient of tbe divisor for tbe coefficient of the 
quotient, and subjoin to it a fraction having for its numerator tbe let- 
ters composing the dividend, and for its denominator those of the 
divisor, and if tbe same letter occurs in both, subtract tbe exponents 
for tbe exponent of tbe same letter in tbe quotient ; if the exponents 
as well as tbe letters are alike, both the quantities may be struck out. 
Tbe same rule applies to tlie signs in division as in multiplication. 

2. To divide a polynomial by a monomial^ divide each term of the 
polynomial by the monomial according to Rule 1, and connect the 
results by their proper signs. 

3. To divide two jwlynomials one by tbe other, arrange them witb 
respect to tbe powers of tbe same letter, then divide tbe first terms 
one by tbe other, and thence will result one term of tbe quotient ; 
multiply tbe divisor by this, and subtract the product from tbe divi- 
dend : proceed witb the remainder in tbe same manner. 

4. To divide algebraic /ractiont^ invert the terms of tbe divisor, 
and proceed as in multiplication. 

Ea!amples, 
1. 'ladH'c^Sab^ y^a*-'b'-'c=^id'bc. 

U. 15 a ♦ b' ^ 5 a- b' = »/ a^-'b'-' = 3 a b\ 

8. 12 «^ — 2a- f -f c -r 2 rt- = a- — c 4- r^ • 

2a^ 

H.r 12j:v l).r//^ 
i (i X -f 12 aw/ — .;? V ^ -t- '\ .v = h --— ^ -^^ = 



CHAP. II.] DIVISION. ,59 

O 

6. Divide af^ -^ Sjp^z -^ Sxsr — sf"* by x ^ z. 
x^z)x^-^SarZ'\-3xz^^z^(a^-'^xz'\-z^ quotient 



— 2ar^2r4- ^x^ 



xz'^z^ 
a? jy^ — 2r* 



Divide a* — b^ by a — b. 

a - 6) a» — 6Va* + a^ 6 + a* 6' + a ft* 4- ft* quotient. 
a^-aH 



a* ft -an* 

a'b''-a'b^ 

Here the second tenn of a^ ft'* 

the dividend is brought c^b^ ^ ab^ 

down to stand over the 



corresponding term in a ft* — ft* 

the last product. aft* — ft ^ 

8. Divide 1 by 1 — a? 

\-^x)\ (1 + 0? + ;r- + ar^ + a^ + ^-^ 

1 —a? ^""^ 



J? — X' 





-«' 


-«« 














*•- 


• ar^ 






ar"* 



60 



INVOLUTION. 



[PAKT I. 



9. 



2d?' 



a' H- X' 
^x 



a -j- X a* -^ x^ 



U X' a + X _ 2 J?- (o + ;p) 
X "~ (a* + 0?*) ;p 



x'^h' 



10. 



X' ■\- hx ^* — 6* 



F.X 



X — b 



x'^^hx'^h'' x--h {x^hf x{x^-h) 



;p' + ft ' ft" 

a? (;r -h ft) (j? — ft) X {x- — ft-) ^ X ' 

11. Divide 96 — 6 a* by 6 — 3a. Qiwt. 16 +8a -j-Aa^ -^Hal 

12. Divide 10a' + 1 1 a'ft — 19 aftc — 15 a^c -(- 3 aft« + ISftc'' 
— 6 b'c by 3 a ft + 5 a- — 5 ft c. Quot, 2 a + ft — 3 c. 

13. Divide x"- +y" + -, by a: -f y + - • QtioL a? — y -f- - . 



Sect. V. Involution, 



1. To invoice or raise monomials to any proposed power. 

Rule, — Involve the coefficient to the power required, for a new 
coefficient. Multiply the index of each letter by the index of the 
required power. Place each product over its respective letter, and prefix 
the coefficient found as above : the result will be the power required. 

All the powers of an affirmative quantity will be -f : of a ne^ive 
quantity, the even powers, as the 2nd, 4th, 6th, &c., will be + ; the 
odd powers, as the 3rd, 5th, 7th, &c., will bo — . 

To involve fractions, apply these rules to both numerator and 
denominator. 

The application of these rules is fully exemplified in the following 
table of 





BOOTS AND POWBBS OP 


U0N0MIAL8. 




Root 


a 


-b 


V* 


e 
a 


a* 
I'x 


2y 
1 


-a- a," 


Square 


<^ 


¥ 


V*' 


a* 


a' 


1 


a'V 


*»«' 


Cube 


a" 


-ft' 


X 


«' 


a" 


8y 
1 


-a"*' 


4th Power 


a* 


V 


X/x' 


a* 


a" 


ley 
1 


a»*'^ 


5th Power 


a" 


-b' 


Vx^ 


a' 




32 y 

1 


- a*^x" 



CHiP. II.] INVOLUTION. 61 

^. To involve polyn&miali, 

M.'Mnldply the given (quantity into itaelf as many times, 
wanting one, as there are units in the index of the required power, 
ud the last product will be the power required. 

Example, 

Cube x±,z and 'Hx — ^z. 

X ±,z 2d? — 3j2r 

2a? — S^r 





x±z 


z-*' 






r±xz 
±xz^ 




QnaTes 


3^±^XZ 

X ±z 


+ Z^ .... 












x'±iix'z 
± x^z 




+ ^ 


Cobes 


\X^±%3?Z 


+ Sxz* 


■hz\ 



4 a?' — Qxz 

— Qxz -\-9z' 

.4:x'''^l2xz + 9z'*- 
fix — 3z 

Sx^^Ux-z-^- 18;r;»* 

— 12«»;2f + 36a?;»« — 27;r^ 



The operation required by the preceding rules, however simple in 
^ nature, becomes tedious when even a binomial is raised to a 
liigii power. In such cases it is usual to employ 

Sir Imoc Newton's Rule /or involving a Binomial. 

1. Tojind the terms without the coefficients. — The index of the first, 
or leading quantity, begins with that of the given power, and de- 
cwisea continually by 1, in every term to the last; and in the 
following quantity the indices of the terms are 0, 1, 2, 3, 4, &c. 

2. To find the uncice or coefficients. — The first is always 1, the 
•Kond is the index of the power : and, for the others, if the coeffi- 
cient of any term be multiplied by the index of its leading quantity, 
ttd the product be divided by the number of t^rms to that place, it 
^giie the coefficient of the term next following*. 

ilToiff.— The whole number of terms will bo one more than the 
Wex of the given power; and when both terms of the root are + , 
ill the terms of the power will be + ; but if the second term of the 
foot he — , all the odd terms will be + , and the CTen terms — . 

* This role, expressed in general terms, is as follows : — 

(• + 4), - «• +».«"-' 6 + n. L=J a-«6« + n . !L±i.'LzJa"-«i«,&c 

2 2 3 

The lame theorem applied to fractional exponents, and with a slight modifi- 
ation, ssrres for the extraction of roots in infinite series ; as will be shown a 
^&rtheron. 



62 INVOLUTION. [part I. 

Eapamples. 

1 . Let a '\- X he involved to the fifth power. 

The terms, without tho coefRcieDts, will he 

a\ a^ Xy d^x\ a^ x\ ax*y x\ 

and the coefficients will he 

^ 5x4 10 X 3 10 X a 5 X 1 

' ^' 2 ' 3 ' 4 ' 5 ' 

and therefore the fifth power is 

a' + ba*x + lOa'x- -h lOa^a?' -f 5aa?' -f jr\ 

Here we have, for the sake of perspicuity, exhibited separately 
the manner of obtaining the several terms and their respective co- 
efficients. But in practice the separation of the two operations is 
inconvenient. The hest way to ohtain the coefficientB is to perform 
the division first, upon either the requisite coefficient or exponent 
(one or other of which may always be divided without a remainder), 
and to multiply the quotient into the other. Thus, the result may 
be obtained at once in a single line, nearly as rapidly as it can be 
written down. 

2. {x ^ yy ^ x'' -\-l a^ z ^ 21 X' z' -\- 35 a?* z' + 35 x^z^ 
-f 21 x'z^ -f 7a?2^ -f z*. 

3. {x ^ zf :=^ 3^ -^^x^ z -\- 28 :i^ ;f*^ - 56 x^ z' + 70 x' z' 

— 50 X' X?^ 4- 28 X' ;?« - 8 4? ;J^ -f ;^^ 

For Trinomials and Quadrinomiah. — Let two of the terms be 
regarded as one^ and the remaining term or terms as the other ; and 
proceed as above. 

Example. 

Involve X -\- y — z to the fourth power. 

Let X be regarded as one term of the binomial, and y — ^ as the 
other : then will U + ^ — zY = {;i? + ( j^ — -2?)}^ = a?* i- 4 ar* (y — z) 
+ 6 a?*^ (^ — z^ -f A: X {y ^ zy + {y — z)\ where the powers of 
y — z being expanded by the same rule, and multiplied into their 
respective factors, we shall at length have a?* -f ^ x^y ^ Ax^ z •\- 
^x'f — l^a^y z + 6aj»ar' + 4 xf — V2xy'z + ^'Hxyz' — 
4 a? ^r* -h y* — 4 y** ^ + 6 / z^ — 4 jy 5r* -f- z\ the fourth power 
required. 

Had {jX -f y) and — z been taken for the two terms of the bino- 
mial, the result would have been the same. 

Note. — The rule for the involution of multinomials is too complex 
to be given in this place. 



CHiP. II.] KVOLUTION. 8.1 

Sbct. VI. Evdutian. 

1 . To find the roots of monomials, 

iSv/e.— Extract the correspondiDg root of the coefficient for the 
Mw coefficient: then multiply the index of the letter or letters hj 
the index of the root, (fractionally expressed, as explained at paee 4,) 
tk result will he the exponents of the letter or letters to he placed 
liter the coefficient for the root required. 

Examples. 
I Find the fourth root of 81 o*;?*. 
First \/ 81 = >/9 = 3, new coefficient. 
Then 4 X -J = 1, exponent of a; and 8 x J = 2, exponent 
of z. 

Hence 3 a :2^ is the root required. 

2. To find the square root of a polynomial. 
Proceed as in the extraction of the square root, in arithmetic, as 
explained st page 29. 

Examples. 
1. Extract the square root of a* + 4 a"* a? + 6 a^ ar^ -f 4 a a?* +ar'. 
a* + 4 a^x -f Qa^x"' -f Aax' + x* {a^ + ^ax + ara* 
tf« [root required. 

4a' a? + 4a=;r^ 



•io' + aaa? 
2 a if 



X- 2a' x^ + 4a;p» - x- 



2. Extract the square root of :r* — S a;^ + * 4?* — i a? + ^^ , 



i 



X 1 

^ 2 ^ 16 

a 1 

' 2 ^ 16 



64 EVOLUTION. [part I. 

3. To find the roots of powers in general. 

When the power (w) of the root to he extracted is not very 
high, the following method may he employed for extracting the nth 
root*. 

Rule. — Range the several terms in the order of the powers of one 
of the unknown quantities ; having done which, extract the nth root 
of the first or leading term, which place in the quotient, and cancel 
such term ; then bring down the second term for a dividend, which 
divide by the term of the root just found, raised to the power of 
(n — 1 ), and multiplied by n, the quotient will be the second term of 
the root. Then to find the third term of the root, involve the two 
already obtained to the nth power as far as the third term, which sub- 
tract from the third term of the given power for a dividend, which 
divided by the same divisor, gives the third term of the required root. 
And, in like manner, to find the fourth term of the root, involve those 
already obtained to the wth power as far as the fourth term, which 
subtracted from the fourth term of the given power, gives a fresh 
dividend ; and thus proceed until the root is extracted. 

Examples. 
1. Extract the cube root of a?« — 6 ar' + 15 a?* — 20 «* + 15 ar* 
— 6a? -h 1. 

^P**- 6a?^+ 15a?*-20af» + 15d?' — 6« + 1 (a?'— -2^?+ 1 
«** [jroat rehired. 



3a?* 



* -6ar^ 

irB-«6«^*-fl2a?* = (a;« — ^f to the third term. 



3a?» 



j,*-_6*» -h 16**~20*"+l5*«-6*+l=(*>-2*+l)* 



2. Find the 4th root of 16a* — 96 0=* a? + ^K^a^x — 216aj?' 
4- 81 x\ 

16a*— 96a^a?-f 216a-a?' — 216a;r'» f 81;i?»(2a-3^ 
16 a* 



8 a"* X 4= 32 a")*- 96 a^^ 



16 g* — 96a-^a? +216o»;r' — 2l6o^-|-81 ** « (2a - 3*V 

* * * * * 



* By this mode of expreision is meant any root whatever; the nile may be 
made to apply to any particuhir case by substituting for n, the power of the root 
required, which in the first Example above is 3, and in the second is 4. 



CHIP. II.] 8UBD8. 05 

ilTdtH.— In the higher roots proceed thus : — 

For the liqiMdraiey extract the tguare root of the square root. 

n nk4 root, ,, cube root of the square root. 

„ ei9M root^ „ sq. rt. of the sq. rt. of the sq. rt. 

n nmtk rooty ,, cube root o£ the cube root. 

Exunples, however, of snch high roots seldom occur in any prac- 
tical inquirieB. 



Sect. VII. Surds. 



A Swrdy or irrational quantity, is a quantity under a radical sign or 
fractioDal index, the root of which cannot he exactly obtained. (See 
Abith. Sect. 9. Evolution.) 

Sards, as well as other quantities, may be considered as either 
ample or compound, the first being monomials^ as v^S, a*, ^a b\ the 
^n poly nomials, as y/3 + v'S, V« + >/*— v'c^* V(«— >/^)' 
^a + 3, &c 

Radons] quantities may be expressed in theybrm of surds, and the 
operation, when effected, often diminishes subsequent labour. 

RBDUCTION. 

1. To reduce eurds into their simplest expressions. 

Ca8B I. If the surd be not fractional, hut consist of integers or in- 
tegral factors under the radical sign : 

Arfe.— Divide the given power by the greatest power, having the 
wne mdex, contained therein, that measures it without remainder; 
let the quotient be affected by the radical sign, and have the root of 
tbe diiieor prefixed as a coefficient, or connected by the sign x . 

Examples, 
1. v^75 = >/(26 X 3) = v'aS X >/3 = 6 ^S. 
a. \/US = X/(U X 7) = V®^ X V = ^ V^. 
8. V176 = V(1«X 11) = V16 X V11 = ^V11- 
4. v^(8af»— 12«*^) = >/4a*(2«-8^)=s >/4«* X 
v^(2a?— 3^) = 2« v'(2«— 3^). 

«. V(56^y + 8^) = VQ^'C^^ -f 1) = V^*^ X 

Cabb II. ^ M« nircf be fractional, it may be reduced to an equiva- 
^mteqralone, thus: — 

. iMe.— Multiply the numerator of the fraction under the radical 
*|n bj that power of its denominator whose exponent is one less 

F 



66 SUBD8. [PABT I. 

than the exponent of the surd, and place it as a whole number under 
the radical sign, and prefix to it for a coefficient, the coefficient of the 
original surd (whether unity or any quantity) divided by the deno- 
minator of the given fraction taken from under the radical sign. 

Note, — This reduction saves the labour of actually dividing by an 
approximated root ; and will often enable the student to value any 
surd expressions by means of a table of roots of integers. 

Ejeamples, 
1. v^^; then I x 3 = 3, and | v'S = v'f 
a. v^|; then 1 x 6 = 5, and |V6 = y/^. 

V- : then 1 x n = n, and - ^n = v^ -. 
71 n n 

VI; then 4 X 5« = 100, and ^ V^^^^ = Vf 



3. 



3 /2 a 



then 2 a X 5a?|* = 50 a«*; and -— VSOaar* 
I 6a! 



6. VM = 2 VA» ^^en 2 X 81»= 13122, and 
^\V13122 = |V18=VM- 

Case TIL If the denominator o/ the fraction be a binomial or r««f- 
dualy of which one or both terms are irrational and roots of%quare% : 

Rule.^ Multiply this fraction by another which shall have its nume- 
rator and denominator alike, and each to contain the same two quan- 
tities as the denominator of the given expression, but connected with 
a different sign. ■ 

Note 1 . — By means of this rule, since any fraction whose nume- 
rator and denominator are the same, is equal to unity ^ the quantity 
to be reduced assumes a new appearance without changing its value ; 
while the expression becomes freed from the surds in the denomma- 
tor, because the product of the sum and difference of two quantities 
is equal to the difference of their squares. 

Examples. 

1 S _ 8 >/5 -f >/3 _ 8(v^5 H- >/3) 

n/5— >/3 >/5— v^3V5 + n/S"" 2 

= 4(^/5 + v^3). 



2. 



3 3 V5- v^2 _3(v^5- y/a) 



v^5 -f >/2 v^5 + ^^2 V5 - ^^2 8 

= >/5 - >/2. 



CHIP. 11.] 9U11M. 67 

V10O-2V6O-I-V36 16-2^^60 ^ 
= 5I3 ^ = ^=8-^^60 = 8-2^15. 

4 >/^^ :^ n/^i^ V5-t/3 ^ V5-V 3 
V5-^V3 V^-^V3'V5-V3 n/5->/3^^ 

Abte2. — Upon the same general principle any binomial or re- 
«doal surd, as V^ — V^ ™*y ^® rendered rational by taking 

VA;-'t v(a— 'B) + V(A"~'BO T V(A""^BO + &c- for a 
Doltipfier : where the upper signs must be taken with the upper, the 
lower with the lower, and the series continued to n terms. 

Thus, the expression ^/o' — \/6', multiplied by \jc^ •\' \/ a h' 

+ V^+ V**> gives the rational product a* — ft\ 

^ To reduce Bwrds having different exponenU to equivalent ones 
that have a common exponent, 

Inrolre the powers reciprocally, according to each other s exponent, 
for Dew powers: and let the product of the exponents be the common 

exponent 

^ofe. — Hence, rational quantities may be reduced to the form of 
ttj isaigned root; and roots with rational coefficients may he so 
Ktiooed as to be brought entirely under the radical sign. 

Examplee. 

1. a"and 6", become a"'" or a"" and ft*"'" or A*"". 

2. a* and 6% become a * ^ or a* and ft* * * or 5* . 

3. si and 2*, become 3* and 2^, or V3' and \/^\ or %/^l 

4. (a + ft)*, and (a - ft)*, become ^\/{a + ft)' and »V(« - *)'• 

5. The rational quantity a', becomes >/a*, V^> ^/®^ ®' V^"*- 

6. 4aV5ft, becomes V(^«)' X V^^ V^^o^x V^^ <»• 

These and other obvious reductions which will at once suggest 
tiiemselTes, being effected, the operations of addition, subtraction, &c. 
ire 80 easily performed tipon such surd quantities as usually occur, 
^ it will suffice merely to present a few examples without 
<J«tailing rules. 

f2 



68 



SURD9. 



[PABT I. 



ADDITION. 

EofA. v'SH- v'18=-s/(4.2)+v'(9.2) = 2v^2 + 3^/2=6>/2. 

2. Add together v^54, ^/^, andv^^. 

^^64 = >/(9 . 6) = v'9 X >/6 = 3 v'e ] The sum of these is 
^/* =n/A=>/Ax >/6 = *n/6 (3-hi + |)^/6 

^/A = n/(V* • i) = >/ « = -J n/6 J = S^lff ^6. 

3. >/27a«ar+ v'Sa'ajrs v'COa* . 3«) + (a^ . 3«) = 
3a»v'3« + a>/3a? = (3a« + a)N/3aT. 

SaVJ + fl" V* = (8« + «')V*• 
8UBTRACTI0N. 

^a?. 1. 2>/50- ^/18 = 2 ^/(25 . 2) - >/(9. 2) = 2 . 6 >/Q 
- 3 v^2 = (10 - 3) V2 = 7 V2. 

4. V^S^^'^ - V10«* « = V(1^5a* . 2a?) - VC^a' • 2«) 
= 6aV2« — 2aV^* = SaV^a?. 

5. >/46**«— v'20«^ar»= >/(9**. 6a?)— v'(4a«ar*. 6*) = 
(3^«2*a?)>/6a?. 



a» - c« / 1 



\ac/ . 



MULTIPLICATION. 

^a?. 1. \/lS X 5 V4 = 6 V(18 . 4) = 6 V(4 .2.9) = 
6 »y (8 . 9) = 6 . 2 V^ = 10 V^- 

2. |^/■^xiVA = f.i^/(i.^Jff) = i^/^^=i^/(^.^) = 
i>/:^(fty = i-AV36 = ^v'36. 

3. a* X a^ = a^"*"i = a'A"*""^ = a+^, 

4. (a? + z)^ X (a? -f ^F = (« + ;?)*"^* = (« + ^)^. 

5. (a? + >/y) X (a? — Vy) = a?» — y. 

6. (« + >/y)* X (ar - ^y)i =^{0?^ y)^. 

'^. jjr" X 3^ = «" *" = j2r "*" . 



CHAf, II.] SURDS. 69 

10. (T X (f^ =€r X -. = flT— . 

a 

11. V— ax >/ — asv'aV— 1 X v'a>/— l=ax— 1=— a. 

12. n/— ax^/ — ft= v'a>/— 1 X v'5>/— l = >/<lft X — 1 

DIVISION. 

iSp.l. VlOOO-5-aV^ = lVT =^V250 = 2V(125.2) 

= 10 V2. 

=«V(A-8) = H.fV8 = MV3. 

8. «i ^ a?* = a?^* =: a?"^ = 1 ^ aj*. 

II II »— II 

4. /^ «- = «•- = «— . 






6. "^^^/(^*-«^)-^-^ ^(a-^) = ^^x 



INVOLUTION. 

&.I. (|a*)«=:|.|.a*+* = ia*=:iV^- 

2- (J^/iy = ^i•i•^/(J•i•i) = :5V^/4 = lV^/(4•|) 

8. (8 + ^/5)« = {(3 + V6) (3 + >/6)} = 14 + 6 ^/5. 
4. (a- v'J)* = «'-3a» >/* + 3a6-6v'*. 

EVOLUTION. 

i&.l. v'lO' = v'lOOO = V(100 . 10) = VIOO X vio 
= 10^/10. 

8. V31aV« = V(91<**y*-5^«) = ^«Wy'^• 
3. ^(a* — 4 a >/5 + ^) = a — 3 ^/5, the operation being 
P^ormed as in the arithmetical extraction of the square root 

Aofe. — The $quare root of a binomial or residual a ± 5, or even 
^ i tiinomial or quadrinomial, may oflten be conveniently ex- 
*neted thus:— Take d = V(a» — ft*); then >/(a ± ft) = 

V ^ ± V ^^' ^"^ "^ evident: for, if /y/^^ ± 



70 SIlfPLS EQUATIONS. [PABT I. 

A / — — be squared, it will give a 4- >/(a* — <^) or a + d, as it 

ought : and, in like manner, the square of a / — a / —5 — * 

is a — >/(a^ — d^)i or a — 6. 

Ex. 1. Find the square root of 3 -j- 2 v'jJ. 
Here a = 3, ft = 2 V^, </ = ^(9 — 8) = 1, 

= Vi + N/f = >/2 + n/1 = 1 + >/2. 

2. Find the square root of 6 -* 2 >/5. 

Here a = 6, ft = 2 >/5, rf == >/(86 - 20) = Vl« = ^ 

3. Find the square root of 6 + >/8 — >/12 — >/24. 

Here a = 6 + ^/8, ft = >/ia -f n/24, rf = v^(6 + >/8)» — 
(n/12 + v^24)-^ = ^/(44 -f 12 ^^8 — 36 - 2 ^^12 . 24) =: 
>/(44 - 36 + 12 n/8 -- 12 -s/8) = v^8. 

o + rf 6 + 2 ^^8 Q.,o A^ — ^ 
Conseq. -y- = ^-^^ = 3 + >/8, and -^ = 

6 + >/8-v/8 . 

2 =^- 

But (Ex. 1), v^(3 -f 2 >/2) =^ ^^(3 -f ^/8) = I -f >/2. 
Therefore the root required is 1 + >/2 — >/8*. 



Sect. VIII. Simple Equations. 

An algebraic equation is an expression by which two quantities, 
called members (whether simple or compound), are indicated to be 
equal to each other, by means of the sign of equality (=) placed be- 
tween them. 

In equations consisting of known and unknown quantities, when the 
unknown quantity is expressed by a simple power, as Xy x\ x\ &c, 
they are called simple equations^ generally ; and particularly, ample 
or pure quadratics^ cubics^ &c. according to the exponent of the nn- 

* For the cube and higher rooto of binomials, &c. the reader may oontult the 
treatises on Algebra, by Madaurin, Emerson, Lacroix, Bonnyoastle, J. R. 
Young, and Bine. 



CliP.U.] 8IMPLI EQUATIONS. 71 

known quantity. But when the unknown quantity appears in two 
or more different powers in the same equation, it is named an ad^ 
/eetorf equation. Thus a;*- = a + 15, is a simple quadratic equation : 
2^ f a f = 6, an adfected quadratic. 

It is the former class of equations that we shall first consider. 

The reduction of an equation consists in so managing its terms, 
tint, at the end of the process, the unknown quantity may stand 
•lone) and in its first power, on one side of the sign =, with the 
ham quantities, whether denoted by letters or figures, on the other. 
Thus, what was previously unknown is now affirmed to be equ4d to 
the aggregate of the terms in the second number of the equation. 

^In general, the unknown quantity is disengaged from the known 
wet, hfpefforming upon both members the revbbsb opbbations,"* 
to those indicated by the equation, whateyer they may be. Thus, 

If any known quantity be found added to the unknown quantity, 
jetitbesabtracted from both members or sides of the equation; or 
if any sacb Quantity be found subtracted, let it be addedf . 

If the nuKnown quantity have a multiplier, let the equation be 
dJTided by it. 

If it be found divided by any quantity, let that become the mul- 
tiplier. 

If any power of the unknown quantity be given, take the corre- 
ipondiug root ; or if any root, find the corresponding power. 

If the unknown quantity be found in the terms of a proportion 
[ir^ Sect. 10), let the respective products of the means and ex- 
^es constitute an equation ; and then apply the general principle, 
ts above. 

Examples. 

1- Given x — 3 + 5=9, to find x. 

Otherwise, in appearance only, not in effect. 

By transposing the 3, and changing its sign, d; + 5 = 9 + 3. 

By transposng the 5, and changing its sign, d; = 9 + 9 — 

*=: 7. 

^ Given 3 « + 5 = 20, to find x. 
First, by transposing the 5, 3 j; =x 20 — 5 «= 15. 
by dividing by 3, j; =s y ss 5. 

* This timple directioii, comprehending the seven or eight particular rules 
^the rednction oi equations given by most writers on algebra, from the time 
*f Ncvtoo down to the present day, is due to Dr. Hutton. It is obviously 
^Knded npon the mathematical axiom, that equal operations performed upon 
ifi*! things produce equal results. 

t These two operations constitute what is usually denominated trantpotUum^ 
^cenae the operation of thus adding or subtracting any quantity from each side 
*f the eqoatiim is moat simply perwnned by tron^poiinf it from one side of the 
■fisiion to Che other, and dianging iu sign. 



72 SIMPLE EQUATIONS. [PAI 

3. Given - -f </= 3 6 — 2<:, to findd?. 

First, transposing dy - = 36 — 2c + </. 
a 

Then, multiplying by a, xs=3ab^^ae-{-ad, 

4. Given V(3* + 4) -f 2 = 6, to find a. 

First, transposing the 2, 51/(^* + ^) = ® ~ ^ = ^• 
Then, cubing, 3« + 4 = 4^* = 64. 
Then, transposing the 4, 3 « = 64 — 4 = 60. 
Lastly, dividing by 3, a? = ^o = 20. 

5. Given 4a^— 56 = 3£^;p + 4tf, to find a. 

First, transposing 5 b and S da^ 4aa; — 3(f;p = 56 + 
Then, by collecting the coefficients, (4 a — Sd)a!^bb -^ 

.'. by dividing by 4 a — 3 </, a? = r r-y 

4 a — ott 

6. Given |a; + j^^ — j^^=3, to find a. 

^f^YS ^ ^*^2 = I 80* + 24x - aO« = 360. 
4 X 6 X 6, we have ) ^ 

That is, collecting the coefficients, 34 a; = 360. 

.-. dividing by 34, a? = ^ = i^ = 10 -J^. 

7. Given ^x : a : : 6 6 : 3 e, to find x. 

Mult means and extremes, ^ c a; s: 5 a 6, 

T%. .,. I o ..1 « 20 a 6 
Dividing byfc, a;=:5ao-ri<? = -^ • 



8. Given a '\- a =^ >/d^ 4- « >/(4 6- + a?'), to find a?. 
First, by squaring, we have, a' + 2 a a? + «' = < 

a? v^(4 6* + a?^) 

Then, striking out a* from both sides, 2 a a? -f- ^ 

X >/(4 6» -i-a:^) 

dividing by a?, 2 a + a? = >/ (4 6* 4- a?*) 

squaring, 4 a' + 4 a a? + a:** = 4 6* + a?^ 

striking out a^ and ) ^, _ , 

transposing 4a%) 

A' M- K . 46^-4a« 6' 

dividing by 4 a, a: = = a. 

4 a a 

9. Given \/ca? — ac = 6 H- y^* — «> to find a?. 



First, dividing by v'^ — «> we have -^/c = —z — — — ^. 



CHi?. n.] BXTBBMINATION. 73 

V'C— I h 



tranqK>8iiigthe 1, ^/c — I, or 



1 ^{x - a) 



mTerting and transpodDg the fractions, — ^—7 -"^sr 

moltiplying by d, >/(« — a) = 



v'c — 1 
sqaanng both sides, 4? — a = — ; — —r- 



iln^.^: 



10. GiTen 18 — >/3 « = >/l3 -f 8 ar, to find a?. 

An$. X = 12. 

8 

11. GiTen y + ^/ 4 + y* = 7/4 . ^8-) > ^ ^°^ y* 

12. GWen J (* + 1) + i (* + ^) == H« + ^) + 10> ^ fi°<l*- 

il9M. d? = 41. 

13. Giyen^^A/^^ s/(a:— 1) :; 3 : 1, to find a:. AnB,\\. 
14 Given (6* + «*)» = (a» + ^)*, to find «. 

EZTBBMINATION. 

Wlen two or more unknown quantities occur in the consideration 
of ID algebraical problem, they are determinable by a series of given 
ixlepeDdent equations. In order, however, that specific and finite 
Miotioos may be obtained, this condition must be observed, that there 
^gwm as many independent equations as there are unknown quan- 
^. For, if tibe number of independent equations be fewer than 
tlie unknown quantities, the question proposed will be susceptible of 
ID indefinite number of solutions'!' : while, on the other hand, a 
greater number of independent equations than of unknown quan- 
tities, indicates the impossibility or the absurdity of the thing at- 
ten^. 

Where two unknown quantities are to be determined from two in- 
^^dent equations, one or other of the following rules may be em- 



^vaknown quantities by two equations ; and so on. 

Thus, if J? + %/y = « -i- >/b 
and a — ^v = c — ^d 
Then #«a,y»6, sr = o, «=«rf. 



74 EXTBRMINATION. [PABT I, 

Bide 1 . — Find the value of oDe of the uDknown letters in each of 
the given equations ; make those two values equal to one another in 
a third equation, and from thence deduce the value of the other un- 
known letter. This substituted for it in either of the former equa- 
tions, will lead to the determination of the first unknown quantity. 

2. Find the value of either of the unknown quantities in one of 
the equations, and substitute this value for it in the other equation : 
80 will the other unknown quantity become known, and then the first, 
as before. 

3. Or, after due reduction when requisite, multiply the first equa- 
tion by the coefficient of one of the unknown quantities in the second 
equation, and the second equation by the coefficient of the same un- 
known quantity in the first equation : then the addition or subtrac- 
tion of the resulting equations (according as the signs of the unknown 
quantity whose coefficients are now made equal, are unlike or like) 
will exterminate that unknown quantity, and lead to the determina- 
tion of the other by former rules. 

Notes. — The third rule is usually the most commodious and expe- 
ditious in practice. 

The same precepts may be applied, mutatis mutandis^ to equations 
comprising three, four, or more unknown quantities : and they often 
serve to depress equations, or reduce them from a higher to a lower 
degree. 

Examples. 

1. Given d^jf* -h 3y = 41, and Sar* — 4y = 12, to find x and y. 
1st equa. x by 3, gives lHaP -f 9y = 123 
2ndequa. x by 4, gives 12 j?* — 16 y = 48. 
The difference of these, 25 y = 75, whence y = 3. 
Then, from 2nd equa., 3 jr» = 12 + 4y = 12 -j- 12 = 24 
Whence dividing by 3, ar* = 8, or a? =: 2. 

Ex. 2. Given a? -h y -r xr = 53, ;i? -f 2y + 3;8f = 105, and 
« + 3y + 4;2r= 134. 

1. X -{- y -h J2r= 53 

2. 0? 4- 2y 4- 34r=: 106 

3. 0? -f 3y + 4;t= 184 

4. 1st equa. taken from 2nd, gives y -f 2« = 52 

5. 2nd equa. taken from 8rd, y -h jp = 29 

6. 5th equa. taken from 4th, jt ss 28 

7. 6th equa. taken from 5th, y = 6 

8. 5th equa. taken from 1st, x = 24. 

Ex, 3. Given x-^yssa, x-^ zssb^ y -\~z^Cy to find Xy y, and s. 

1. ar -f-y = a 

2. X -^z= b 

3. 1/ -^ z^si c 



(IIP. II.] OSNUUL PROBLEMS. 75 

4. lft + 2nd + 3rd, gives 2ar + 2y + '^^^==0+ i + c. 

5. Ha]f 4th equa. gives ^-ry + ^ = ia + 4^ + i^- 

6. 3rd equa. taken from 5th, gives ^=|a-f|5— Ic. 

7. dud eqna. taken from 6tfa, ^ = 1^ — 1^ + 1 c 

8. 1st eqoa. taken from 6tfa, Jtrss _|a + i^ -\- ic. 
Ex, 4. Given ax -\-by^ e^ and o^^p -f ^'y as ^, to find 47 and y. 

cV — h<f , atf — ca^ 

&.6. Given ajp -f ^y + ^Jjzrassrf, c^x + V y -rf^ z ^ df of' x + 
5"^ + c^'ar =: rf" to find a?, y, and ;2r. 

^""a^V - ac'6" -f CO' ft'' - 6aV' + hifa^' - c5V 

'"aftV' — ac'ft" +ca'ft"-fta'c" + ftc'a"-cA'a''' 

& 6. Given d:(;p -h y + ^) = 18] 

y (^ + y -r ^) = 27 I to find a?, y, and z. 
^(^-f^ + 4 = 86 J 

iln«. :i?=:2, y = 8, j2raB4. 

^.7. Given (a? + y)- « 60, and(ay+y)- = 2^, to finda?andy. 

y ^ 

Ans. ^ = 10, y = 2. 
^. 8. Given 4* + 4y + i;? = 62] 

l^+|yH-i^ = 47[ to find a?, y, and 4?. 

ilw*. a? = 24, y = 60, ;2r = 120. 
SOLUTION OP GENERAL PROBLEKS. 

A general algebraic problem is that in which all the qaantities 
^^Bconed, both known and unknown, are expressed by letters, or 
^general characters. Not only such problems as have their 
ttnditions proposed in general terms are here implied ; but every 
^^f^cdar numeral problem may be made general^ by substituting 
^'^ for the known quantities concerned in it ; when this is done, 
^ problem which was originally proposed in a particular form 
^^^^taes general. 

In solving a problem algebraically, some letter of the alphabet must 
w mbstituted for an unknown quantity. And if there be more 
QBbown quantities than one, the second, third, &c., must either be 
^^piwied by means of their dependence upon the first and one or 
<W9 of the data conjointly, or by so many distinct letters. Thus, 
^>Hay separate equations will be obtained, the resolution of which. 



76 OBNBRAL PB0BLBM8. [PABT I, 

by some of the foregoing rules, will lead to the determination of the 
quantities required. 

ExampU$. 

1. Given the sum of two magnitudes, and the difference of their 
squares, to find those magnitudes separately. 

Let the given sum be denoted by «, the difference of the squares 
by d; and let the two magnitudes be represented by x and y 
respectively. 

Then, Uie first condition of the problem expressed algebraically 
is « + y = *. 

And the second is oj' — ^ = d. 

Equa. 2 divided by equa. I, gives x^y=i- 
Equa. 1 added to equa. 8, gives 2 a; = - + « = 



Equa. 4 divided by 2, gives x = 

Equa. 5 taken from equa. 1, gives y^zs^ 



9 8 



%8 
«* + D «•— D 



^8 ^8 

To apply this general solution to a particular example, suppose the 

sum to be 6, and the difference of the squares 12. Then » = 6 

and D= 12, 

«« + D 86 + 12 48 , 
and X = — - — = — -- — = — = 4. 
2« 12 12 

«* — D 86-12 24 ^ 
and y = — r^ — = — rr — = —- = 2. 
^ 2« 12 12 

Suppose, again, 8 = 6, d = 5 : 

. 25+6 ^ . 26-6 ^ 

then X = — Jq— = 8, and y = — Jq" "^ ^ 

Ex. 2. Given the product of two numbers, and their quotient, to 
find the numbers. 

Let the given product be represented by/?, the quotient by ^; and 
the required numbers by x and y, as before. 

Then we have, 1. xy^=^p^ 

X 

and 2. - = ^. 
V 



Equa. 2 X by y, gives, xssqy 

Substituting this value) ^^ ^^ 

of X for it in equa. 1 J ^^ ^ 

P 
Dividing by ^, y* = — 

Extracting the square root, y = a/^ 



CHI?. 11.] QUADRATICS. 77 

Then, \j snbstitation, a? = yy = ^ ^ /?. = ^ /?^ = ^//^y• 

Suppose the prodnct were 50 and the qaotient 2. 

Theny=^^=^^= V26 = 5, and «= V/?y = 
^/lOO = 10. 
Agun, rappoee the product 36, and the qaotient 2| . 

Then y = y^| = ^^ =^/16 = 4,anda^=V;>y=: 

V81 = 9. 

Ex. 3. OiTen the sum (<) of two numbers, and the sum of their 
aqoues s, to find those numbers. 
iw. « = J * + J y^ 2 8 — ««, and^ = J « — J v^2 s — **. 

£x. 4. The sum and product of two numbers are equal, and if to 
eitber snm or product the sum of the squares be added, the result 
inll be 13. What are the numbers l—Ans. each = 2. 

Ex.5, The square of the greater of two numbers multiplied into 
& leaB, produces 75 ; and the square of the less multiplied into the 
prater produces 45. What are the numbers ? 

Bx, 6. A man has six sons whose successive ages differ by four 
Jem, and the eldest is thrice as old as the youngest. Required their 
•emalages?— iliw. 10, 14, 18, 23, 26, and 80 years. 



Sect. IX. Quadratic Equaiionx. 

QuiDRATic Equations are such as contain the square of the unknown 
^UDtitj, and which, after due redaction, may be made to assume the 
Seseral form Adr' + B« + <' = 0; then dividing by a, the coefficient 

^the first term, there results «^ -|- - « H- - = 0, or, making pss- ^ 



A A 



9 = -, we have 

a!» + jD« + y = (1) 

*n equation which may represent all those of the second degree, 
J^Vidg being known numbers, either positive or ne^tive. 

Ut a be a number or quantity which, when substituted for ^, rcn- 
fe8^+/?a?-f ^ = 0; thenar ^ pa -f ^ = 0, or y== -^a^^pa. 
^Wqaently j?' + jt> « + y, is the same thing as;i^ — a*+J^^~" 
^«, or as (« + a) (« — a) -^ p (a ■— a), or, lastly, as (a? — a) 
(«+a + ;?). 

The inqairy, then, is reduced to this, viz. to find all the values of 
* which shall render the product of the above two factors equal to 
*^g. This will evidently be the case when either of the factors 



78 QUADRATIdS. [?ART I. 

is = ; but in no other case. Hence, we have x-^Oy ^0, and 
« -h a -f J^ = 0, or ;i? s=s a, and a? s= — a — p*. 

And hence we may conclude — 

1. That every equation of the second degree whose conditions are 
satisfied by one value (a) of a?, admits also of another value ( — a — p). 
These values are called the roots of the quadratic equation. 

2. The sum of the two roots a and — a — p is =^ — p; their pro- 
duct is — a^ — apy which as appears above is = g. So that the eoefi- 
cienty jD, of the second term is the sum of the roots with a contrary 
sign ; and the known term^ q^ is their product, 

3. It is easy to constitute a quadratic equation whose roots shall 
be any given quantities b and d. It is evidently a^ — (6 -|- rf) 
X -k-hdzzz 0. 

4. The determination of the roots of the proposed equation (1) is 
equivalent to the finding two numbers whose sum is — jp, and pro- 
duct q. 

5. If the roots b and d are equal, then the factors x — b and x^d 
are equal ; and ar •\' p x -^ q \s the square of one of them. 

To solve a quadratic equation of the form a? •\' px ^ ^ =: 0, let it 
be considered that the square of ^ + | j9 is a trinomial, a^ + /> a? + \jfy 
of which the first two terms agree with the first two terms of the 
given equation, or with the first member of that equation when q is 
transposed. 

That is, with aj^ + jt? a? = — y. 

Let then \f^he added, we have 
0^ -{- px -^-^p^ ^ \p^ — q 
of which the first member is a complete square. 

Its root is a? -f ^jt? = ± >/ {\p^ — q) 

and consequently iP = — i j» ± 'J{\j^ — q) 
otherwise, from number 2 above, we have 
X -\- of := — p and xaf =^q. 

Taking 4 times the second of these equations from the square of 
the first, there remains a^ ^^xaf + ^=jp* — 4j' 

Whence, by taking the root, a? — a?' = >/(/?* — 4 y) 

Half this added to half equa. 1, gives 

And the same taken from half equa. 1, gives 

which two values of a evidently agree with the preceding. 

It would be easy to analyze the several cases which may arise, ac-^ 
cording to the different signs and different values, of/? and q. Bat 

* If it be affirmed that the given equation admitN of another valae of jt^ 
besides the above, h for instance, it may be proved as before that # — 6 mnst 
be of the number of the factors of «■ + p * + 9, or of (* — a) (* + fl + ji). 
But » — a and jr -f a -f p being prime to each other, or having no oommon 
factor, their product cannot have any other factor than they. Conieiiuently 6 
must either be equal toaorto — a— p; and the number of roots is restricted 
to two. 



CHIP. II.] 




QUADRATICS. 




79 


tliese need not here 


be 


traced. It is evident that whether there he 


flTCB 


1. 
2. 
3. 
4. 


^ + />* = q 

3^ ^ px == q 

3^ + px = — 

3^ -- px = — 


9 





The general method of solution is by completing the square^ that 
ii, adding the square of |/>, to both members of the equation, and 
tlien extracting the root. 

It may fariber be obsenred that all equations in which there are 
two tenns inTolving the unknown quantity or any ftinction of it, and 
tbe index of one double that of the other, may be solved as qua- 
dntics, by completing the square. Thus : 

It M 

^ + pi» = ^, «*■ i />«* = y, a?^ ± /> a?^ = y, {jf^ -^ p X -\- qf 
±(^ -{-px ^ q) =zr,{af^ -^ ary ± (aj*" — j^) = y, &c., are 
of tbe same form as quadratics, and admit of a like determination of 
die unknown quantity. Many equations, also, in which more than 
one unknown quantity are invoWed, may be reduced to lower dimen- 
Boni by completing the square and reducing ; such, for example, as 

(!? + /)» ± /» (:2r^ + y) = ^, 3 ± ^ = ^, and so on. 

Sr y 
N(fte, — In some cases a quadratic equation may be conveniently 
nhed without dividing by the coefficient of the square, and thus 
vithoQt introducing fractions. To solve the general equation a o^ ± 
*«=<;, for example, multiply the whole by 4 a, whence 4 a* a?® di 
iahx = 4 a <;, adding ^ to complete the square, 4ta^aP ± iab x 
+ ft' = 4 a'c -f P taking the square root, 2 a « ± J = di 

*J(iae + ^); whence x = ^ — ^ }" -: which will 

Krre for a genera] theorem. 

Examples, 

I Giren x^ — 8 x + 10 = 19, to find x, 
transnosing the 10, dj» — 8a? = 19 — 10 = 9 
completing the square, a?^ — 8« + 16s= 9-fl6 = 26 
extracting the root, a? — 4 = ± 6 
consequently a? = 4±:5 = 9or — 1. 

^ Given -^ — ^ -— , to find the values of x, 

X ST 9 

multiplying by «*, 10 a? — 14 + 2 a? = — — , 

tnnsposing, ^ «* — 12a? = — 14, 

dividing by V, ««-^«=-^, 

cmplet. squ. x» - 44 oi + (^^ = iH - « = -i¥r> 

extract, root, « — 4i » ± -fr, 

tmnspottng, x^^±^^S or ^. 



80 QUADRATICS. [PARI I. 

3. Given «*-f2a? + 4 >/«* -h 2 a? 4- 1 = 44, to find ». 
adding 1, we have (j;* -|- 2a: + 1) + 4 ^{x^ + 24? 4- 1) =45 
complet squ. (a:* -f 2a! + 1) 4- 4 >/(jj* + 2a? -f + ^ = 49 
extract, root, ^{^ -|-2a?H-l)-H2 = ±7 

transposing the 2, ^{o^ -f 2a? +1) = ±7 — 2 = 5 or — 9 
that is,a?4-l = 5 or — 9 
hence a? = 4 or — 10. 

4. Given a?" — 2 aa?^ = c, to find x. 

n 

complet. squ. a;" — 2aa?^ -|-a?=:c-fo^ 
extract, root, a:^ — a = ±: >/ (c + a*) 

n 

transposing, a?^ = a di ^^^ (c + o^) 



consequently, x =^ {a '±. y/c -f a*)". 

•E^ 4;r 
6. Given ^4- — = 12, and a? — y = 2, to find a? and y. 

tr y 

3^ X 

complet. squ. in equa. 1, — -f4-4-4=16. 

Extracting root - -f 2 = ±: 4 : whence - = 2 or — 6, and 

a?= 2y or — 6y. 

Suhstituting the former value of x in the 2nd equa., it becomes 
2^ — y = 2, or y = 2 ; whence a? = 4. 

Again, substituting the 2nd value of a;, in equa. 2, it becomes 
— 6y — y or — 7y = 2; whence y = — ^, and a? = + V . 

6. Given a?*y' — 5 = 4 a? ^, and i a? y = ^ y*, to find a? and y, 
equa. 1, by transposition, becomes a?*y* — 4a?y = 6 
completing the square, a^}^ — 4a?y + 4 = 9 
extracting the root, a?y — 2 == ± 3 
whence a?y = 5 or — 1. 
Substituting the first of these values for a?j^ in equa. 2, it becomes 
|. y = ^ : whence ^ = 1 and a? = 5. 

Substituting the 2nd value in the same equation, it becomes 
I y = — J : whence y = — \j\ = — i V^^> *°^ a? = — 1 

ilfw. a? = ± 3, or ± >/^. 

8. A man travelled 105 miles at a uniform rate, and then found 
that if he had not travelled so fast by two miles an hour, he would 
have been six hours longer in performing the same journey. How 
many miles did he travel per hour ? 

Am. 7 miles per hour. 



CHAP. II.] EQUi^TTONS. 81 

9. Find two socb numbers ttmt the sum, product, and difference 
of their squares may be equal. 

Ans. \ + i >/5, and ^ + 5 >/5. 

10. A waterman wbo can row eleven miles an hour with tlie tide, 
ud two miles an hour against it, rows five miles up a river and back 
sgaJD in three hours : now, supposing the tide to run uniformly the 
same way during these tbree hoars, it is required to find its velocity ? 

Ans. 4J^ miles per hour. 



Sect. X. Equations in General. 

Equahons in general may be prepared or constituted by the 
maltipiication of factors, as we have shown in quadratics. Thus, 
snppose the values of the unknown quantity x in any equation were 
to be expressed by a, b, c, </, &c., that is, let a; = a, a? = 6, ^ = c, 
-p=(i, &c., disjunctively, then will x — a=:0, ar — ft = 0, ar — <?=0, 
*-rf=0, &c., be the simple radical equations of which those of 
tiie higher orders are composed. Then, as the product of any two 
of th^ gives a gvadratic equation, so the product of any three of 
them, as (4? — a) (4J — b) {x — c) =: 0, will give a cubic equation, 
or one of three dimensions. And the product of four of them will 
coosdtute a biquadratic equation^ or one of four dimensions ; and so 
OD. Therefore, in general, the highest dimension of the unknown 
fiOHHtff X is equal to the number of simple equations that are mul- 
fiplied together to produce it. 

When any equation equivalent to this biquadratic (x — a) (jx — b) 
(* t) {x — d) = is proposed to be resolved, the whole difficulty 
wniists in finding the simple equations a? — a = 0, x — 6 = 0, 
Jf-c=0, X — ^ = 0, by whose multiplication it is produced; for 
each of these simple equations gives one of the values of x^ and one 
wlotion of the proposed equation. For, if any of the values of x 
<ledoccd from those simple equations be substituted in the proposed 
equation, in place of x^ then all the terms of that equation will 
^iih, and the whole be found equal to nothing. Because when it 
i« sopposed that ^ =: a, or x^b^ or ;p = c, or x z= dy then the 
product (4? — a) (x — b) (a? — c) (a? — d) vanishes, because one 
of the factors is equal to nothing. There arc therefore four supposi- 
tions that give (x — a) {x — b) {x — c) {x — <^) = 0, according 
to the proposed equation ; that is, there are four roots of the pro- 
posed equation. And after the same manner any other eauation 
•dmita of as many solutions as there are simple equations multiplied 
^ one another that produce it, or as many as there are units in the 
behest exponent of the unknown quantity in the proposed equation. 
Bnt as there are no other quantities whatsoever besides these four 
K ^ c, d^) that, substituted in the proposed product in the place of 
^ wfll make that product vanish ; therefore, the equation {x — a) 
(*-h){x — #r) (;p — df) = 0, cannot possibly have more than these 



82 EQUATIONS. [part 

four roots, and cannot admit of more solutions than four. If ^ 
substitute in that product a quantity neither equal to a, nor 6, nor 
nor dy which suppose e, then since neither e — a, c— i, e — c, n 
e — dyis equal to nothing; their product cannot be equal to nothir 
but must be some real product : and, therefore, there is no suppo 
tion beside one of the aforesaid four, that gives a just value of 
according to the proposed equation. So that it can have no mc 
than these four roots. And after the same manner it appears, i\ 
no equation can have more roots than it contains dimensions of i 
unknown quantity. 

To make all this still plainer by an example, in numbers, suppc 
the equation to be resolved to be x^ — 10 ar* + 35 ;ir — 50 j? + 24 = 
and that we discover that this equation is the same with the prodi 
of (a? — 1) (j?— Q) {x — 3) {x— 4), then we certainly infer tl 
the four values of a: are 1, 2, 3, 4 ; seeing any of these numbe 
placed for Xy makes that product, and consequently a?' — 10 dr* 
35 x^ — 50ii? + 24, equal to nothing, according to the propos 
equation. And it is certain that there can be no other values of 
besides these four : for when we substitute any other number for 
in those factors ii? — 1, d? — 2, x — 3, ;r — 4, none of them vanL 
and therefore their product cannot be equal to nothing, accoi'ding 
the equation. 

A vai'iety of rules, some of them very ingenious, for the soluti 
of equations, may be found in the best writers on Algebra*; 1 
we shall simply exhibit the easy rule of Trial-and-Error, as it is giv 
by Dr. Hutton^ in the 1st vol. of his ** Course of Mathematics." 

Rule for the general solution of Equations hy Trial-and-Error, 

" 1 . Find, by trial, two numbers, as near the true root as possib 
and substitute them in the given equation instead of the unknot 
quantity ; marking the errors which arise from each of them. 

" 2. Multiply the difference of the two numbers, found by tri 
by the least error, and divide the product by the difference of t 
errors, when they are alike, but by their sum when they are nnlil 
Or say, as the difference or sum of the errors is to the differei 
of the two numbers, so is the least error to the* correction of 
supposed number. 

'^ 3. Add the quotient, last found, to the number belonging to t 
least error, when that number is too little, but subtract it when t 
great, and the result will give the true root nearly, 

" 4. Take this root and the nearest of the two former, or a 
other that may be found nearer ; and, by proceeding in like mann 
a root will be had still nearer than before ; and so on to any degi 
of exactness required. 

" Note — It is best to employ always two assumed numbers tl 
shall differ from each other only by unity in the last figure on I 
right; because then, the difference, or multiplier, is 1." 

* i'^ee the trentiseii of I/acroix, BonnycMtle, Wriod, J. R. Vming, &c. 



CBIMI.] 



BQUATIONS. 



83 



Example, 

To find the root of the eohic equation ^r* + ^v' + x = 1 00, or the 
kloe of X in it. 



Here it is soon found that x 
L^ between 4 and 5. Assume, 
b^refore, these two numbers, 
M^ the operation will be as fol- 



iSup. 

le 

64 

84 

-16 



2nd Sup. 



X 



. sums 
. errors 



5 
25 
125 

155 

-^55 



the som of which is 71. 

TbcnasTl : 1 :: 16 : -225. 
Hence x = 4*225 nearly. 



Again, suppose 4*2 and 4*3, 
and repeat the work as fol- 
lows : 



\9tSup. 

4-2 

17-64 

74088 

95-928 

—4072 



X 

x' 
x' 



2nd Sup. 

4-3 

18-49 

79-507 



sums . . 102-297 



errors . . +2* 297 



the sum of which is 6*369. 

As 6-369 : 1 :: 2297 : 0036 
This taken from . . 4*300 

leaves x nearly = 4*264 



Agiio, soppose 4-264 and 4*265, and work as follows : 



4-264 
18'181696 
7:'526752 

W-972448 

-0-027552 



X 

x^ 
x" 

sums 



4-265 
18-190225 
77-581310 

100036535 

+ 0-036535 



the sum of which is •064087. 

Then as 064087 : -001 : : -027552 : 0-0004299 
To this adding ... 4*264 



gives X very nearly = 4-2644299 



When one of the roots of an equation has been thus found, then 
^t for a dividend the given equation with the known term trans- 
ited to the unknown side, so as to make the equation equal to 
•olkiog ; and for a divisor take x minus the root just determined : 
(he quotient will be equal to nothing, and will be a new equation 

o 2 



84 PR0OBE8SION. [PABT 

depressed a degree lower than the former. From this a new Tat 
of X may be found : and so on, till the equation is reduced to a q%: 
dratic, of which the roots may be found by the proper rules. 

Example, 

Given the biauadroHc equation, «* — 1 1 «^ -f 28 «^ -f 36 ^e 
144 = 0, to find the four roots. 

First, by the above method of Trial-and-Error, we find one oF 
roots to be 6 ; then, 

a!-.6)a?«— lla?'-f 28a?^ + 36a; — U4(a?^ - 5 «' - 2*4- 
a?«— 6 a?' 



— 


6a?' 


-f-28a?' 
+ 30a?^ 


+ 








-.2 a?'' 
-2 a?* 


36 a? 
12 a? 



24 a? — 144 
24 a?— 144 



The quotient of which, ar' — 6 a?*' — 2 a? -f 24 = 0, is a eMc ^^ 
tion of which one of the roots is found to be 4 ; then, 

-r - 4 )ar» - 6 a?* — 2 a? + 24 (a?^ — a? — 

ar»-4a?'* 



a?' — 2a? 

a?' + 4a? 

— 6a? + 24 

— 6a? + 24 



The quotient of which, a?^ — a? — 6 = 0, is a quadratic equation, 
which the two roots are readily found to be 3, and — 2 ; thus t J 
original biquadratic equation is composed of the factors (a? — 
(a? - 4), (a? - 3), and (a? + 2). Or, 

(a? + 2).(a?-3).(a?-4).(a?-6) = a?»--l]a?'»+28a^+36a?~144=t 



Sect. XI. Progression. 
When a series of terms proceed according to an assignable orde 
either from less to greater or from greater to less, by continual eqn 



<* 



CSIP. II.] ARITHMETICAL PB0GRBS8I0N. So 

diflerences or by saccessive equal products or quotients, they are said 
to fonn a/iro^retttois. 

If tbe quantities proceed by successive equi-differences they are 
«id to be in Arithmetical Progression, But if they proceed in the 
«iDe continued proportion, or by equal multiplications or divisions, 
ibev are said to be in Geometrical Progression. 

If the terms of a progression successively increase, it is called an 
^Kotiing progression : if they successively decrease, it is called a 
^ktcmding progression. Thus, 

1, 3, 5, 7, 9, &c. form an ascending arithmetical * "^ 
^2^ 20, 18, 1 6, &c. form a descending arithmetical 



> Progrcssi 



iion. 



1, 3, 9, 27, 81, &c. form an ascending geometrical 
^ ^ 1) !» 49 ^^' ^'^1'°^ A descending geometrical / 

ABITHMBTICAL PBOOBESSION. 

1 Let a be the first term of an arithmetical progression, 

d the common difference of the terms, 

z the last term, 

n the number of terms, 

9 the sum of all the terms. 
Tbcnei,a-f(i; a-^-'Stdj a-^-^d, &c., is an ascending progres-sion, 

and a, a^dj a — Sd!, a^^dj &c., a descending progression. 
Hence, in an ascending progression, a 4- (« — 1) <^ ^^ the last term ; 

m a descending progression, a — (n — 1 ) </, is the last term. 

^Lcta series be a+{a + d) -f (a +2 </) + (« + 3 rf). 

TlKwne inverted (a + 3rf) -f (« + 2</) + (« + <if) + « 

^«amofthetwo(2a-i-3rf)+(2a-f3<ir)4-(2a+8<Q-f(2a+8rf)=2*. 

Thitii, (2a + 3</) x 4, in this case (a -f a + 3rf)n = 2#. 

. Cowe^ntly, »=J(« + a"+"37)n, or = i(<» + ^)»»> "°<^ * 
>kie = a -f 3</. The same would be obtained, if the progression 
*^ descending; and let the number of terms be what it may. 

3. From the equations ;? = a -f (« — 1) <^ * = 4 »* (<• + ^)» ^^^ 

'=*}«{a -^0 + (n-- !)<'}» ^6 ™*y readily deduce the following 

!^^<Kemi applicable to ascending series. When the series is descend- 

^ other U)e signs of the terms affected with d must be changed, 

I g I *<BiQ8t be taken for z; and vice versd. 



« 2« 

/9\j * — « «* — fl^ ^zn — ^s 2# — 2na 

»— 1 2« — ;» — a n» — n n» — » 

(3.)* = a + nrf-d=* +i*rf- Jrf = 



86 OKOMETRICAL PROGRESSION. [pART I. 

(4.) «=|w(a + ;2r) = (a-f ^«(/-|rf)n=:(z-^«</ + ^d) n 

" Ud 

(5.) w = = ; I. 

^ ^ a -^ z d 

Examples. 

1. Required the sum of 20 terms of the progression 1, 3, 5, 
7, 9, &c. 

Here a =: 1 , </ = *^, w = 20 ; which heing suhstitutcd in the 
theorem « = (a + ^7«(/ — | </) 7«, it hecomes « = (1 -f 20— 1) '20 
= 20 X Hi) = 400, the sum required. 

Note, — In any other case the sum of a series of odd numbers 
beginning with unity, would be = w^, the square of the number 
of terms. 

2. The first term of an arithmetical progression is 5, the last 
term 41, the sum 299 Required the number of the terms, and the 
common difference. 

^« 59H ,^ , , ^ 

Here n = = — - =: 13, the number of tenns, 

a -I- 5r 46 

and d = = — -- — = 3, the common difference. 

w — 1 12 

3. There are 8 equidifferent numbers : the least is 4, the greatest 
82. What are the numbers ? 

TT * z -^ a 32 — 4 ^ , ,. _ 

Here d = 7 = — z — = 4, the common difference. 

n — 1 i 

Whence 4, 8, 12, 16, 20, 24, 28, 32, are the numbers. 

4. The first term of an arithmetical progression is 3, the number 
of terms 50, the sum of the progression 2600. Required the last 
term and the common difference. 

Here ;2r = — — o = — ; 3 = 104 — 3 = 101, the last term, 

n 50 

, ^ ;2r — a 101 — 3 ^ . 

and d = = — = 2, the common difference. 

w — 1 49 

5. The sum of six numbers in arithmetical progression is 48 ; and 
if the common difference d be multiplied into the less extreme, the 
product equals the number of terms. — Required those terms. 

Ans. 3, 5, 7, 9, 11, and 13. 

GEOMETRICAL PROGRESSION. 

Let a be the first term of a geometrical series ; 
r the common ratio ; 
z the last term ; 
n the number of terms ; 
s the sum of all the terms. 



CHAP. II.] 



GEOMETRICAL PROGRESSION. 



«: 



. am: 






tea 






Then a, ra» r'o, r* a, r"""'a, is a geometrical progression, 

whicli will be tucending or descending^ according as r is an integer 
or zfraaioH, 

Let the prog, a -f ra -f r*a + r^a -f r*a = «, be x by r, it 
becomes ra -^ f^a -{- r^ a -{- r^a •\- r^a ^:^rs. 

The diff. of these is, — a -^^ r^a^^ rs — ». 

But r^a is the last term of the original progression multiplied 
by r, or in general terms r"""' a x r^ that is r" a. Consequently 
r*<i — a = r* — ». 

«T, r"a — a r" — I , ^ , 

whence « = = —a ^ the sum of the series. 

r — 1 r — 1 

A amikr method will lead to a like expression for «, whatever be 
the nlue of fi. If r be a fraction, the expression becomes trans- 

lormed to # = a. 

1 — r 

Now from these values of z and s the following theorems may be 
deduced. 









; * -f r;2r — r*. 



r r" — 1 



w*= 



j(r» — 1) _ q(l — r") _ r^j2_l— ^^^ ^^ 



1 -r 



r— 1 



r" — r- 



(4.)r 






(5.) ji = log- y 4- log. ^-- log- « 
log. r 

And, if the logarithm of - = N, that of = M, and 

tbtofrsR: then 

Also, if when r is a fraction, n is infinite, then is r" = 0, and the 
^ipKttion for • becomes 

(7.) 9 ss , which expression is often of use in the summa- 

1 — r 
tion of infinite series. 



h The least of ten terms in geometrical progression is 1, the 
'^io 2. Required the greatest term, and the sum. 



88 FRACTIONAL AND NEGATIVE EXPONENTS. [PART 1. 

Here z = ai^~^ = 1 x ii'* = 512, the greatest term ; 

rz —a 2 X 512 — 1 

and « =: = = 1023, the sum. 

r — 1 1 

2. Find the sum of 12 terms of the progression 1, ^, ^, j^*-^, &c. 
Hei^ , = ^_, = _^pL =^^^^, the sum. 

3. Find the sum of the series 1, |, ^, |, &c., carried to infinity. 
Here by Theor. (7.) % = , becomes * = j = ^> ^^*® 

sum required. 

4. Find the vulgar fraction equivalent to the circulating decimal 
•36363636. 

This decimal, expressed in the form of a series, is, -^^^^ -f- T;^Jrt,) 
+ TOoVoort + &c., where a = -^^^ , and r = -^J^ . 

Consequently, a = = -f^^ -r tVo = i^' '^*® fraction sought. 

5. Find the sum of the descending infinite series 1 — x -r jr 

- J7* -f X\ &C. 

Here a = 1, r = — ;u, and s = = ; , the sum req. 

And, by way of proof, it will be found that if 1 be divided by 
I + ^, the quotient will be the above scries. 

6. Of four numbers in geometrical progression tbc product of the 
two least is 8, and of the two greatest 128. What are the numbers? 

AnB. 2, 4, 8« and 16. 



Sect. XII. Fractional and Negative Exponents. 

In the preceding sections on powers and surds, we have only 
spoken of such powers as have positive and integral quantities for 
their exponents, and which are termed direct powers. And we ex- 
plained that the exponent of the quantity, expressed the number of 
times that that quantity was multiplied by itself; so that to denote 
the division of any power by its root, we have only to subtract 

a" 
1 from tbe index or exponent of that power ; thus a" * = — ; and 

a" 

by continuing to divide by a, we have a'~'"* = a**"* = — ^, and 

a* a* 

a""^"' = a""' = — ; or generally, a"~* =— ^; that is, to express 

the division of one power of a qiiantitt/ by any other potter of the 
same quantity ^ we subtract the exponent of the divisor from that of the 
dividend. 



I 



eUiP. II.] FRACTIONAL AND N£OATIVK BXP0NBNT8. 89 

If ft b greater than m, then will the quotient -^ = a'' have for 

i K^ exponent {p) a positive numher, and is then called a direct power 
^-.*ta'y bat if m is greater than n, p will he negaUive^ and a"' is then 

a* 
^r^led an inverse power of ay thus --j = a*""* = «*, is a direct 

a* 
^>«wer, but — j- = a'~* = a~^ is an inverse power. 

Now, to arrive at a just idea of the value of a quantity with a 

x^^gadye exponent or of an inverse power, let us successively subtract 

mmwityfrom tlie index of any direct power; or, in other words, divide 

t.liftt power successively by its root ; thus, let a" be the direct nS^ 

pcwerofo, then 





«— ' = 


a' 

a' 




a»-' = 


T-- 




a»-' = 


^ = . = . 




«'-' = 


°'='='. 



a a 

1 1 1 1 

a <r 

"* a? • "* a* "^ ' 

^, generally, tf~* = — . TheU is^ the value of an inverse potg>er 

Septal to unitff divided by the same direct power. 

As the properties of inverse powers are the same as those of direct 
Jowen, all that has been stated in Section V. regarding the latter, 
^oally applies to the former, and it is therefore unnecessary to 
recapitulate it here. 

We have, however, yet to mention another kind of exponent, viz., 
the fractional, used to express the roots of quantiUes, in a similar 
manner to that employed for expressing powers ; and which extends 
to them also, all the properties already stated as belonging to inverse 
tnd direct powers. 

This method of expressing any root of a quantity, consists in 
ittaching to that quantity as an exponent, a fraction having unity for 
its numerator, and the index of the root for its denominator. 



i 



90 LOGARITHMS. [pABT I. 

Thus \/a is written or 
1 

i_ 

And the roots thus expressed may be employed in algebraical cal- 
culations in the same manner as powers with integral exponents. 
Thus, if we wish to express the square of the cube of jr, we do so by 
x'^ ^ - = a/* ; and in like manner, to express the square of the cube 
root of ;r, we should have j?l><* =: x'; or, the square of the square 

root of X =. A^^ z:^ xi =^ X itself, which is obvious. And in 
It 

general, x" may be employed to express the n^ power of the ni^^ 

root of J*, or what is the same thing, the m^^ root of the n*^ power 

of X. 

If the fractional exponent has a negative sign, its value will be 

found by the rule already given for integral inverse powers; thus, 

-" 1 1 

«'" = — = ;;;— r^ = Unity dividcd bv the m}^ root of the w'** 

power of a. 

It only remains to state, that in using fractional exponents, we 
may substitute any equivalent fraction, or a decimal, in place of the 

original exponent; thus a-, a*, a**, a*^, are only so many different 
ways of expressing the square root of a; advantage may frequently 
be taken of this circumstance, to facilitate the working with surds. 

By way of recapitulation, then, there are four different kinds of 
exponents employed in analytical operations, viz. : — 

The posittce iyitegral exponent, as .r", which denotes the direct nth 
power of a?, and is equal to x multiplied w times by itself. 

The ne(fative integral exponent, as a; " ", which denotes the inverse 
n^^ power of ar, and is equal to unity divided n times by x. 

The positive fractional exponent, as «", which denotes the direct 
n*** root of ;r, and is equal to a quantity, which being multiplied 
n times into itself will equal x. 

The negative fractional exponent, as x~ » , which denotes the in- 
verse n^^ root of a;, and is equal to unity divided by the direct 
n^ root of X, 



Sect. XIII. Logaritfnns. 

LooARiTHiis are a series of numbers in arithmetical progression, 
answering to another series in geometrical progression ; so ti^en that 
in the former corresponds with 1 in the latter. 



CHAP. II.] LOGARITHMS. 91 

Thus, 0, 1, 2, 3, 4, 5, are the Logs, or aritkmetieai series ; 

and 1, 2, 4, 8, 16, 32, | ^'^ ^^^ ^^f ' ^' 9e(nnetricd series, an- 
' ' ' ' ' 'J swenng thereto. 

Or, 0, I, 2, 3, 4, 5, the logarithms ; 

and J, 5, 25, 125, 025, 3125, the corresponding numbers. 

Or, 0, 1, 2, 3, 4, 5, the logarithms ; 

and 1, 10, 100, 1000, 10000, 100000, the corresponding numbers. 

In which it will be seen, that by altering the common ratio of the 
geometrical series, the same arithmetical series may be made to serve 
as the logarithms of any series of numbers. As above, where the 
common ratios of the several geometrical series are 2, 5, and 10, re- 
spectively. 

Or, the logarithms of a number may be considered as the indices 
or exponents of the powers of some root or radix, which when in- 
volved to the power indicated by the logarithm, is then equal to the 
number. Thus, taking the foregoing series as an illustration, in 
which the roots or radices are 2, 5, and 10, respectively, we have 

Nos. Logs, Nos. Logs. Nos. JLog». 

1 = 2 « i = 5 " 1 = 10 " 

2 = 2' 5 = 6* 10 = 10 * 
4 = 2 =* 25 = 5 * 100 = 10 « 
8 = 2'* 126 = 5 •'* 1000 = 10 » 

16 = 2 ^ 625 = 5 ^ 10000 = 10 * 

32 = 2 * 3126 = 5 * 1(M)000 = 10 * 

in which the exponents in the columns headed Xo^«., are the logar- 
rithms of the numbers in the first column headed Nos. 

The last of these series, or that in which the common ratio of 
the geometrical series, or (what is the same thing) the radix of the 
system of logarithms is 1 0, is that usually employed for the purpose 
of facilitating calculations ; and is the same as the series of logarithms 
contained in the Logarithmic Tables in the Appendix to this work. 

As in this system of logarithms the numbers whose logarithms are 
integral increase as the powers of 10, and it is requisite in calcula- 
tions to use the logarithms of the intermediate numbers, it becomes 
necessary to interpolate with fractional logarithms ; thus the series 
then becomes 



Nos. Logi. 


Nos. Loot. 


1 = 10 •• 


8 = 10 •*>** 


2 = 10 •«"« 


9 = 10 •«*«» 


3 = 10 *"^'"" 


10 = 10 »• 


4 = 10 •«»• 


11 = 10 «•«««« 


6 = 10 «■«»»' 


12 = 10 »•»»•« 


6 = 10 *"»»^» 


IS = 10 ••'«»" 


7 = 10 "^^^ 


&c. &c. 



in which the exponents of the powers of 10, answering to any given 
uamber, will be found to correspond with the logarithm of that num- 
ber as given in Table H. in the Appendix. 



92 LOGARITHMS. [PABT I. 

In order to explain the properties oF logarithms, and the reason of 
the rules given in the section on Logarithmic Arithmetic, let as as- 
sume any series of numhers, N, N^, Ng, N^, &c., to which corre- 
spond the logarithms, /, l^y Ly l^y &c., to the same root or radix (r). 

1. Then we have, N = r*, Nj = r'*, N^ = r\ &c. ; and by the 
rule for multiplpng powers, (page 66,) r'xr'» = r'^'>=Nx N,; 
and N.Nj .Ng =r'+'» + '«. Or the logarithm of the product oj 
two or more numbers^ is equal to the sum of the logarithms of those 
numbers; and conversefyy the logarithm of the quotient of two num^ 
berSj is equal to the difference of the loaarithms of the dividend and 
divisor ; which correspond with the rules given at page 45, for the 
multiplication and division of numbers by logarithms. 

2. IfN = Nj =No = N3; then we have, by the foregoins, 
log (N.N.N .N)= log (N') = r'+ '+' + ' = r*'; and in genenU, 
log (N") = r"'. Or the logarithm of the n** power of any quaniitj/ is 
equal to n times the logarWim of thai quantity^ whether n is int^al, 

fractional^ negative^ or positive. 

For let A be the quantity, and let n be negative ; then A"" = — ;, 

and log ( — ) = ^ogof I ^ n (log A) ; and as log of 1 =: 0, 

log 1 — » log A = — » (log A) = log ( A-"). 

P - 

If n = — ; then, let A ? = K ; raising both sides to the power ^, 

we have A'' = K^, and their logarithms, p (log A) = ^ (log K) ; then 

P - 

dividing both sides by 7, we obtain - (log A) = log K = log (Al). 

p -t I L 

Lastly, if » = — -, then will A « = — ; and since log (A •) =a 

^ a5 

|(log A) .-. log/i \= logl -^ (logA)= - ^ (logA) = 

log(A--0- 

8. In Section XI., page 41, it was stated that the decimal portion 
of a loearithm was always the same for the same digits, and that ii 
was only the characteristic which was altered, by the multiplication 
or division of the quantitv by 10, or any power of 10 ; the reason of 
this is obvious, since all the powers of 10 have integers only foi 
their logarithms. 

4. It was also stated, (page 41,) that it was only the characteristic 
of the logarithm of a firaction, which was negative, the decimal 
portion being always positive; to illustrate this, let us take the 

fraction -05 = -^ = — -j— = lO"* x lO**"*^** = 10 - 2+ -cwwo . 

and therefore 2-698970 is the log. of -05. 

5. Suppose there be two systems of logarithms whose roots or 



CHIP. II.] LOGARITHMS. 93 

bises are r and «. Let any number N have p for its loearithm in the 
first system, and q for its logarithm in the second : we shidl have N = r^ 

and N = I* ; which gives »* = #», and * = r». Therefore, taking 

the logarithms for the system r, we shall have log 9 =^(logr); or, 

if in the system r we have log r = 1, then log » = -, or y = 

pi 
l£- =s p X . . Thus, knowing the logarithm p of any number 

N, for the system whose base is r, we may obtain the logarithm q of 
tlie same number for the system «, by multiplying je? by a fraction 
vlioee numerator is unity and denominator the logarithm of 8 taken 
Id the sjstem r. 

6 In the system of logarithms first constructed by Baron Napier, 
the great inventor, 9 = 2*718281828459, &c., and the exponents 
ire usually denominated Napierian^ or Hyperbolic logarithms; the 
latter name beins given because of the relation between these log- 
triihms and the lines and asymptotic spaces in the equilateral hyper- 
bola: 80 that in this system n is always the hyperbolic logarithm of 
(2-718*28, &c.)*. But in the system constructed by Mr. Briggs 
(correspondm^ with the spaces in a hyperbola whose asymptotes 
make an angle of 25° 44' 25" 28'"), called common or Briggean 
logarithms, r = 10 ; so that the common logarithm of any number 
is, as already stated, the index of that power of 10 which is equal to 
the said number. 

7. Although the Briggean logarithms are those usually employed 
for the purpose of computation, the hyperbolic are always used in 
the differentia] calculus, and the higher branches of analysis ; it there- 
fore becomes somedmes requisite to find the hyperbolic logarithm of 
a quantity, which may be obtained from the common logarithm as 

follows : from § 5, above, we have q =: p -. ; then, * putting 

^' log for hyperbolic logarithm, log for common logarithm, and sub- 

«itating for * its proper value, we have h. log n = log n -. ^717757^^7 

log 2'71o281 

= log n Q.^3^^g^^Q = 2-3025851 (log «) ; or io obtain the hyper- 

Ik^Hc logarithm of any number^ multiply its conmion logarithm by 
2-S025851. 

In practice the following method will be found more convenient 
tlam multiplying by 2-3026851. 

To convert common logs, into hyperbolic. 
Write the common log, as shown in the following Examples, and then 
^e from Table I. the equivalent value of each figure 10 hyperbolic 
1^ taking care that the latter are each moved as many places to the 
'ight as the corresponding numbers in the common logarithm. The 
8nm of the whole will then be the hyperbolic logarithm required. 



1)4 



LOGARITHMS. 



[part I. 



To convert hypei-hdic logs, into common. 

Proceed in the same manner, only using Table II. instead of 
Table I. 



TABLE I. 


TABLE XL 


S: «yp-^^»- 


Hyp. 
Logs. 


Com. Logs. 


1- 2-3025851 

2- 4-6051702 

3- , 6-9077553 

4- 9 2103404 

5- 1 11-5129255 

6- ' 13-8155106 

7- 161180957 

8- , 18-4206807 

9- 1 20-7232658 


1 

2- 
3- 
4- 
5- 
6- 
7- 
8- 
9^ 


•4342945 
•8685890 
1-30-28834 
1-7371779 
21714724 
2-6057669 
3-0400614 
3-4743559 
3-9086503 



Examples. 
I. What is the hyp. log. of 1662 ? 

By reference to Table III. in the Appendix, we find that 3-220631 
is the common log. of 1662 ; then 



7-415778? 



Cora. Log. 




Hyp. Log. 




3-000000 


= 


6-907755 


3 


•200000 


= 


-460517 





•020000 


= 


•046051 


7 


-000600 


= 


•001381 


5 


-000030 


= 


-000069 





•000001 


log. ( 


•000002 
7-415778 


3 


3-220631 




i common 


)f the numb 


er 


Hyp. Log. 




Com. Log 


, 


7-000000 


^ 


3-040061 


4 


•400000 


= 


-173717 


7 


•010000 


= 


•004342 


9 


-005000 


^ 


•002171 


4 


•000700 


^ 


-000304 





-000070 


= 


•000030 


4 


-000008 


= 


•000003 


4 


7-416778 


3220631 





CHAP. II.] COMPUTATION OP PORMULiG. 95 

Sect. XIV. ComptUcUion of FormulcB. 
Since the comprehension, and the numerical computation of for- 
iiu]« expressed aJgebraically, are of the utmost consequence to prac- 
ical men, enabling them to avail themselves advantageously of the 
Lbcorelical results of men of science, as well as to express in scien- 
tific language the results of their own experimental or other re- 
semrthes; it has appeared expedient to present brief treatises of 
Aiitbinetic and Algebra. The thorough understanding of these two 
initiatory departments of science will serve essentially in the applica- 
QOD of all that follows in the present volume ; and that application 
mar probably be fieusilitated by a few examples, as helow : — 

£i. I. Let 6 = 5, <r = 12, d = 13, and s = ; then what 

Kthe numerical value of the expression \/« («— ^) • (* — c) . (« — </) 
from page 1 27, which denotes the area of the triangle whose sides are 
a, 12, and 18? 

„ 6-hc + J 5 + 12 + 13 ,^,, . 

Here i = — ^^-^^ — = — ■ — ^ = 15 ; then « — ^ = 15 

-5 = 5; <~c= 15 — 12 = 3; and« — </= 15 — 13 = 2. 

CoDseqaently, by substituting the numerical values of the several 
qoMtities between the parentheses for them, we shall have 

v/(l5 X 10 X 3 X 2) = x/900 = 30, the value required. 

The same values being given to 6, c, and c?, we may, as a verifica- 
tiOQ of the above, compute the area of the same triangle from the 

eqaiTalent expression, /\J ^ - ( ^^ "^ 2/ ' 2* 

Herer = 12- = 144 ; 6- = 5- = 25 ; 2rf = 26; and ^ = 6|. 
Sabstituung these, the expression becomes 

the same result as before. 

^J?. 2. Suppose ^ = 32^, < = 6 : required the value of J g f, 
•D expression denoting the space in feet which a heavy body would 
Wl TerticaUy from quiescence in six seconds, in the latitude of 
I<ondon. 

Here |^^ = 16-^^ x 6* = 96i x 6 = 579 feet. 

^3. Given D = 6, rf = 4, A = 12, a = 3*141593 ; required the 
»iloeof y^ «^A(D' + D</ + d% a theorem for the solid content of 
* conic fhwtam whose diameters of the two ends are D, </, and 

lieigbtA. 



9fi COMPUTATION OF FORMULA. [PART I. 

Here D- = 36, D r/ = 6 x 4 = 24, d' =16,^ = -2618 nearly. 

Hence -jV v A (D* + D<f + d') = -2618 (36 + 24 -f 16) 12 = 
•2618 X 76 X 12 = 3141593 x 76 = 238-761068. 

Ex, 4. Let a = I, A = 25, ^ = 193 inches : what is the value of 
2 a ^g h ? This heing the expression for the cuhic inches of water 
discharged in a second, from an orifice whose area is a, and depth 
below the upper surface of water in the vessel, or reservoir, A, both 
in inches. 

Here2av^^A=:2 v^(25 x 193) = 10 >/193 = 10 x 13-89244 
= 138*9244 cubic inches. 

Ex, 5. Suppose the velocity of the wind to be known in miles per 
hour ; required short approximative expressions for the yards per 
minute, and for the feet per second. 

First 1760 ^ 60 = *y^ = 29 J = 30 nearly. 

Also 5280 -h (60 x 60) = f^jo = gg = 4^ = 1 J nearly. 

If, therefore, n denote the number of miles per hour : 

30 n will express the yards per minute; and 1|«, the feet per 
second. 

These are approximative results : to render them correct, where 
complete accuracy is required, subtract from each result its 45th part, 
or the Ji/lh part of its ninth part. 

Thus, suppose the wind blows at the rate of 20 miles per hour : 
Tlien 30 n = 30 X 20 = 600 yards per minute, or more cor- 
rectly 600 — Y:^ = 600 — 13i = 6865 yards. 
Also 1 J w = 30 feet per second ; 
or, correctly 30 — ^ J = 30 — ^ = 29^ feet 

Conversely, f of the feet per second will indicate the miles per 
hour, correct within the 45th part, which is to be added to obtain the 
true result. 

Ex. 6. To find a theorem by means of which it may be ascertained 
when a general law exists, and what that law is. 

Suppose, for example, it were required to determine the law 
which prevailed between the resistances of bodies moving in the air 
and other resisting media, and the velocities with which they move. 
Let V,, V.^, denote any two velocities, and R,, R.,, the corresponding 
resistances experienced by a body moving with tliose velocities : we 
wish to ascertain what power of V , it is to which R j is propor- 
tional. Let X denote the index or exponent of the power : then will 
V* : Vlj : : Rj : R^, if a law subsist. 



CHAP. II.] COMPUTATION OF FORMULiE. 97 

(V \' R 

— I : : 1 : — ^ . 

(Vo\' R 
— ^ I = — ^. This, expressed logarithmically, gives 

* X log^=log^; 

^^^^logR^_-_logR, 
logV, - logV/ 

Hence the quotient of the difTerences of the logs, of the resistances, 
divided hy the difference of the corresponding velocities, virill express 
the exponent z required. 

This theorem is of very frequent application in reference to the 
motion of cannon balls, of barges on canals, of carriages on rail- 
roads, &c., and may indeed be applied to the planetary motions. 

When two or more values of any quantities occur in a formula, it 
is usual to denote both of them by the same letter, distinguishing 
between them by the attachment of a small number below it, as in 
the last example above, where the two velocities are both denoted by 
the letter V, but distinguished by the numerals, as Vj and V^, and 
the two resistances by R^ and R^. 



08 OKOMETRICAL DBF1NITI0N8. [PART I. 



CHAP. III. 

PLANE AND SOLID GEOMETRY. 

Sect. I. Defi?iitions. 

I. Geometry is that departmcDt of science, by means of which 
we (Icnionstrate the properties, affections, and measures of all sorts of 
ma^7iitude, 

ii. Magnitude is a term used to denote the extension of any thing, 
and is of three kinds ; as the magnitude of a /twe, which is only in 
one direction, viz. length ; the magnitude of a surface^ which is in tvo 
directions, viz. length and breadth; and the magnitude of a solid^ 
which is in three directions, viz. lengthy breadth and depth, 

3. A ]X)int has no parts or magnitude ; neither length, breadth, nor 
thickness, and serves only to assign position, 

4. A line has length without breadtli or thickness, and indicates 
direction and distance. 

Cor, The extremities of a line are, therefore, points. 

5. A right line is that which lies evenly, or in the same direction, 
between two points. A curve liyie continually changes its direction. 

Cor, Hence there can only be one species of right lines, but there 
is an infinite variety in the species of curves. 

6. An angle is the inclination of two lines to one another, meeting 
in a point, called the angular point When it is formed by two rig^ 
lines^ it is a plane angle, as A ; if by curve lines, it is a curvUineal 
angle, as B. (Fig. 1.) 

7. A right angle is that which is made by one right line A B fall- 
ing upon another C D, and making the angles on each side equal, 
that is, A B C = A B D ; so that the line A B does not incline more 
to one side than another : it is then said to be perpendicular to the 
line C D. All other angles are called oblique angles. (Fig. ^.) 

8. An obtuse angk \s greater than a right angle, as R. (Fig. 3.) 
0. An acute angle is less than a right angle, as S. (Fig. 3.) 

10. Contiguous or adjacent angles^ are the two angles formed by 
one line falling upon another, as R, S. (Fig. 3.) 

I I . Vertical or opposite angles, are those made on contrary sides 
of two lines intersecting one another, as A £ C, D £ B. (Fig. 4.) 

12. A surface has length and breadth, but no thickness, and shows 
extension. The area of a figure is the quantity of space which its 
surface occupies. 



CHAP. III.] ANGLES, AND BIGHT LINES. 99 

Cor, The boundaries or limits of a surface are lines. 

13. A plane is that surface which lies perfectly even between its 
extremes; or which, being cut by another plane in any direction, its 
section would be a straight line. 

14. A idid is a magnitude extended every way, or which has 
length, breadth, and depth. 

Cor, The boundaries or extremes of a solid are surfaces. 

15. The square of a right line is the space included by four right 
lines equal to it, set perpendicular to one another. 

16. The rectangle of two lines is the space included by four lines 
equal to them, set perpendicular to one another, the opposite ones 
being equal. 

17. One right line is said to he parallel io another, when both lines, 
being extended to any length, will never meet, but always preserve 
the same distance between them. 



Sect. II. Of Angles^ and Right Lines^ and their Rectangles, 

Prop. I. If to any point C in a right line A B, several other right 
lines D C, £ C are drawn on the same side ; all the angles formed at 
the point C, taken together, are equal to two right angles ; thus, 
ACD + DCE + ECB = the two right angles A C P + P C B. 
(F%. 5.) rEuclid, Book I. Prop. 13.] 

Cor, 1. AH the angles made about one point in a plane, being 
taken together, are equal to four right angles. 

Cor. 2. If all the angles at C, on one side of the line A B, are 
found to be equal to two right ancles ; then A C B is a straight line. 
(Fig. 5.) [Euc. B. I. Prop. 14.J 

II. If two right lines, A B, CD, cut one another, the opposite 
angles C £ B and A £ D will be e^ual. (Fig. 4.) [Euc. B. I. Prop. 15.] 

III. A right line, B I, which is perpendicular to one of two pa- 
rallels, is perpendicular to the other. (Fig. 6.) 

IV. If a right line C G, intersects two parallels AD, F H ; the 
alternate angles, ABE, and B £ H, will be equal. (Fig. 6.) QEuc. 
B. I. Prop. 29.] 

Cor. 1. The two internal angles D B £ and B E H on the same side 
are equal to two right angles. (Fig. 6.) 

Cot. d. The external angle C B D, is equal to the internal angle on 
the same side B £ H. ^Fig. 6.) [Euc. B. I. Prop. 28.] 

V. Right lines, parallel to the same right line, are parallel to one 
another. [Euc. B. I. Prop. 80.] 

VI. If a right line A G be divided into two parts A B, B C ; the 
square of the whole line is equal to the squares of both the parts, and 
twice the rectangle of the parts ; or A C* = A B* + B C'^ + 2 A B x 
BC. (Pig. 7.) [Euc. B. II. Prop. 4.] 

VII. The square of the difference of two lines A C, B C, is equal 
to the sum of their squares, wanting twice their rectangle ; or A B^ ^ 
AC" + BC»— 2AC X BC. OrAB* -f 2AC X BC = AC* -f 
B C*. (Fig. 7.) [Euc. B. II. Prop. 7.] 

H 2 



100 TRIANGLES. [PABT I. 

VIII. The square of the sum of two lines is equal to the sum of 
their squares, together with their rectangle. 

IX. The rectangle of the sum and difference of two Hoes is equal 
to the difference of their squares. 

X. The square of the sura, together with the square of the differ- 
ence of two lines, is equal to twice the sum of their squares. 



Sect. III. Of Triangles. 

DEFINITIONS. 

1 . A triangle is a plane figure bounded by three right lines^ called 
the sides of the triangle. 

2. An equilateral triangle is one in which all the three sides are 
equal. 

3. An isosceles triangle has only two sides equal. 

4. A scalene triangle has all its three sides unequal. 

5. An equiangular triangle is one which has three equal angles; 
and two triangles are said to be equiangular, when the angles in the 
one are respectively equal to those in the other. 

6. A right-anghd triangle is that which has one of its angles a right 
angle. The side opposite to the right angle is called the hypotenuse^ 
and the other two sides the legs. 

7. An oUique angled triangle has all its angles oblique. 

8. An obtuse angled triangle has one of its angles obtuse. 

9. An acute angled triangle has all its angles acute. 

10. In a triangle, the lowest side, as A B, is called the hase^ and 
the opposite angle C the vertex : the altitude of a triangle is the per- 
pendicular height (CD) from the base, or its extension to the yertex. 
(Fig. 8.) 

1 1. Similar triangles are those whose angles or sides are Tesped^ 
ively equal, each to each. And homologous sides are those lying 
between equal angles. 

PROPOSITIONS. 

Prop. I. In any triangle ABC, if one side B C be produced or 
drawn out ; the external angle A C D will be equal to the two inter- 
nal opposite angles ABC and BA C. (Fig. 9.) [Euc. B.I. Prop. 32.1 

II. In any triangle, the sum of the three internal angles is equal 
to two right angles. [Euc. B. I. Prop. 32.] 

Cor, 1. If two angles in one triangle be equal to two angles in 
another : the third will also be equal to the third. 

Cor, 2. If one angle of a triangle be a right angle, the sum of the 
other two will be equal to a right angle. 

III. The angles at the base of an isosceles triangle, are equal. 
[Eiic. B. I. Prop. 5.] 

Cor, 1 . An equilateral triangle is also equiangular ; and the con- 
trary. 

Cor. 2. The line which is perpendicular to the base of an iaosceles 
triangle, bisects both it and the yertical angle. 



CHIP, in.] TRIANGLES. 101 

IV. In any triaogle, the greatest side is opposite to the greatest 
ugle, and the least to the least. [Euc. B. I. Prop. 18.] 

V. In any triangle ABC, the sura of any two sides B A, A C, is 
greater than the third B C, and their difTerence is less than the third 
ade. (Fig. 10.) [Euc. B. I. Prop. 20.] 

VI. If two triangles ABC, a 6 c, have two sides, and the included 
togle equal in each ; these triangles, and their correspondent parts, 
M be equal. (Fig. 11.) [Euc. B. I. Prop. 4.] 

VII. If two triangles ABC and abc^ have two angles and an in- 
cloded side equal, each to each ; the remaining parts shall he equal, 
•ndthe whole triangles equal. (Fig. 11.) [Euc. B. I. Prop. 26.] 

VIII. Triangles of equal bases and heights are equal. [Euc. B. I. 
Prop. 37.] 

IX. Triangles of the same height, are in proportion to one another 
as their bases. [Euc. B. VI. Prop. 1.] 

X. If a line D E be drawn parallel to one side B C, of a triangle ; 
the segments of the other sides will be proportional ; that is, 
AD:DB:: AE : EC. (Fig. 12.) [Euc. B. VI. Prop. 2.] 

Cor, 1. If the segments be proportional, A D : D B : : A E : EC ; 
then the line D £ is parallel to the side B C. (Fig. 12.) 

C<fr. 2. If several lines be drawn parallel to one side of a triangle, 
>I] the segments will be proportional. 

Cor. 3. A line drawn parallel to any side of a triangle, cuts off a 
triiDgle similar to the whole. 

XI. Id similar triangles, the homologous sides are proportional ; 
thatis,AB : AC :: DE : DF. (Fig. 13.) [Euc. B. VI. Prop. 4.] 

XII. Like triangles are in the duplicate ratio, or as the squares of, 
their homologous sides. [Euc. B. VI. Prop. 19 ] 

XIII. In any triangle, the difference of the squares of the two legs 
A Band A C, is equal to twice the rectangle contained by the base 
Be, and the distance DO, of its middle point from the perpendicu- 
lar DA. (Fig. 14.) 

XIV. In a right-angled triangle B A C, if a perpendicular be let 
fill from the right angle upon the hypothenuse, it will divide it into 
two triangles, similar to one another and to the whole, A B D, A D C. 
(rig.l4.) [Euc. B. VI. Prop. 8.] 

Cor, 1. The rectangle of the hypothenuse and either segment is 
eqoal to the square of the adjoining side. 

XV. The distance A O of the right angle, from the middle of the 
hjpotbenuse is equal to half the hypothenuse. (Fig. 14.) 

XVI. In a right-angled triangle, the square of the hypothenuse is 
eqwd to the sum of the squares of the two sides. [Euc. B. I. Prop. 47.] 

XVII. If the square of one side of a triangle be equal to the 
•om of the squares of the other two sides ; then the angle compre- 
^ed by them is a right angle. [Euc. B. I. Prop. 48.] 

XVIII. I fan angle A, of a triangle B A C be bisected by a right 
be A D, which cuts the base ; the segments of the base will be 
Proportional to the adjoining sides of the triangle; that is, 
BD: DC :: AB : AC. (Fig. 10.) [Euc. B. VI. Prop. 3.] 



102 QUADRILATERALS AND POLYGONS. [PABT I. 

XIX. If the sides be as the segments of the base, the line A D 
bisects the angle A. (Fig. 10.) [Euc. B. VI. Prop. A.l 

XX. Three lines drauTi from the three angles of a triangle to the 
middle of the opposite sides, all meet in one point. 

XXI. Three perpendicular lines erected on the middle of the three 
sides of any triangle, all meet in one point. 

XXII. The point of intersection of the three perpendiculars, will 
be equally distant from the three angles ; or, it will be the centre of 
the circumscribing circle. 

XXIII. Three perpendiculars drawn from the three angles of a 
triangle, upon the opposite sides, all meet in one point. 

XXIV. Three lines bisecting the tliree angles of a triangle, all 
meet in one point. 

XXV. If D be any point in the base of a scalene triangle, ABC: 
then is A B- X D C -f A C- x B D = AD^ xBC-fBCxBD 

X DC. (Fig. 10.) 



Sect. IV. Of QuadrUatercdB and Pdi/gons, 

DEFINITIONS. 

1. A quadrangle or quadrilaieral^ is a plane figure bounded by four 
right lines. 

2. A paraUelogram is a quadrangle whose opposite sides are pa- 
rallel, as AGBH. The line A B drawn to the opposite corners is 
called the diameter or diagonal. And if two lines be drawn parallel 
to the two sides, through any point of the diagonal, they divide it 
into several others, and then C and D are called parallelograms about 
the diameter; and E and F the complements; and the figure £ DF a 
gnomon, (Fig. 15.) 

3. A rectangle is a parallelogram whose sides are perpendicular to 
one another. 

4. A square is a rectangle of four equal sides and four equal 
angles. 

5. A rhombus is a parallelogram, whose sides are equal, and all its 
angles oblique. 

6. A rhomboid is a parallelogram in which only the opposite sides 
arc equal, and all its angles oblique. 

7. A trapezoid is a quadrangle, having only two sides parallel, as 
Fig. IG. 

8. A trapeziuyn is a quadrangle that has no two sides parallel, 
as Fig. 17. 

9. A pdygon is a plane figure enclosed by many right lines. If 
all the sides and angles are equal, it is called a regular polygon, and 
denominated according to the number of sides or angles, as a pen- 
tagon having five sides, a hexagon^ having six sides, a heptagon^ having 
seven sides, &c. 

10. The diagonal of a quadrangle or polygon is a line drawn be- 
tween any two opposite comers of the figure, as A B. (Fig. 15.) 



diP.ni.] QUADBI LATERALS AND POLYGONS. 103 

11. The height of a figure is a line drawn from its vertex^ perpen- 
^licular to the base* or opposite side on which it stands. 

12. Like or nmiiar figures, are those whose several angles are 
•equal to one another, and the sides about the equal angles proper- 
lioQiL 

13. Homelogmte sides of two like figures are those between two 
angles, respectively equal. 

14. The perimeter or circumference of a figure, is the compass of 
it, or sum of all the lines that enclose it. 

15. The internal angles of a figure are those on the inside, made 
l»y the lines that bound the figure, as A B C, D C B, &c. (Fig. 18.) 

16. The external angles of a figure are the angles made by each 
lide of a figure, and the adjoining side drawn out, as B A F, A £ G. 
(Fig. 18.) 

PROPOSITIONS. 

Pbop. I. In any parallelogram the opposite sides and angles are 
equal ; and the diagonal divides it into two equal triangles. [Euc. 
B. I. Prop. 34.] 

II. The diagonals of a parallelogram intersect each other in the 
middle point of both. 

III. Any line B C passing through the middle of the diagonal of 
» parallelogram P, divides the area into two equal parts. (Fig. 19.) 

IV. Any right line B C drawn through the middle point P of the 
diigooal of a parallelogram, is bisected in that point ; or B P = P C. 

V. In any parallelogram A G H B, the complements £ and F are 
equal. (Fig. 15,) [Euc. B. I. Prop. 43. J 

VI. Parallelograms of equal bases and heights are equal. [Euc. 
B. I. Prop. 36.] 

VII. The area of a parallelogram is double the area of a triangle 
binngthe same base and height. [Euc. B. I. Prop. 41.] 

VIII. Parallelograms of the same height are to one another as 
tbeir bases. [Euc. B. VI. Prop. 1.] 

IX. Parallelograms of equal bases are as their heights. 

X. Parallelograms are to one another, as their bases and heights. 

XI. In any parallelogram the sum of the squares of the diagonals 
ii eqoal to the sum of the squares of all the four sides. 

XII. The sum of the four internal angles of any quadrilateral 
%ore, is equal to four right angles. 

XIII. If two angles of a quadrangle be right angles, the sum of 
tlie other two amounts to two right angles. 

XIV. The sum of all the internal angles of a polygon is equal to 
tvice as many risht angles, abating four, as the polygon has sides. 

Cor. Hence all right-lined figures of the same number of sides, 
hate the sum of all the internal angles equal. 

Xy. The sum of the external angles of any polygon is equal to 
four right angles. 

Cor, All right-lined figures have the sum of their external angles 
eqoal. 



104 CIRCLBP, ETC. [part I. 

XVI. In two similar figures AC, PR; if two lines BE, Q T, be 
drawn after a like manner, as suppose, to make the angle C B E = 
R Q T ; then these lines liave the same proportion as any two 
homologous sides of the figure ; viz., 

BE:QT::BC:QR::AB:PQ::AD:PS. (Fig. 20.) 

XVII. All similar figures are to one another as the squares of 
their homologous sides. 

XVIII. Any figure described on the hypothemise of a right- 
angled triangle, is equal to two similar figures described the same 
way upon the two legs; that is, BFC=ALC -fAGB. (Fig. 21.) 
[Euc. B. VI. Prop. 31.] 

XIX. Any regular figure ABCDE, is equal to a triangle whose 
base is the perimeter A B C D E A ; and height the line O P, drawn 
from the centre, perpendicular to one side. (Fi<r. 22.) 

XX. Only three sorts of regular figures can fill up a plane surface, 
that is, the whole space round an assumed point, and these are six 
triangles, four squares, or three hexagons. 



Sect. V. Of the Circle^ and Inscribed and Circumscribed Figures, 

DEFINITIONS. 

1. A circle is a plane figure described by a right line moving 
about a fixed ]>oint, as A C about C : or it is a figure bounded by a 
curved line, every part of which is equidistant from a fixed point. 
(Fig. 23.) 

2. The centre of a circle is the fixed point about which the line 
moves, as C. (Fig. 23.) 

3. The radius is the line that describes the circle, as C A. (Fig. 23.) 
Cor, All the radii of a circle are equal. 

4. The circumference is the line described by the extreme end of 
the moving line, as A B D E A. (Fig. 23.) 

5. The diameter is a line drawn through the centre, from one side 
to the other, as A D. (Fig. 23.) 

6. A semicircle is half the circle, cut off by the diameter, as ABD. 

7. A quadrant,, or quarter of a circle, is the part between two 
radii perpendicular to one another, as C DE. (Fig. 23.) 

8. An arc is any part of the circumference, as A B. (Fig. 24.) 

.9. A sector is a part bounded by two radii, and the arc between 
them, as A C B. (Fig. 24.) 

10. A chord is a right line drawn through the circle, as D F. 

1 1. A segment is a part cut off by a right line, or chord, as D £ F, 
or DABfI (Fig. 24.) 

12. Angle at the centre is that whose angular point is at the centre 
A C B. (Fig. 24.) 

13. Angle at the circumference is when the angular point is in the 
circumference, as BAD, or BCD. (Fig. 25.) 

14. Angle in a segment^ is the angle made by two lines drawn 



CHAP. III.] CIRCLES, ETC. 105 

irom some point of tbe arc of that segment to the ends of the base ; 
IS BCD is an angle in the segment BCD. (Fig. 25.) 

15. Angle upon a segment is the angle made in the opposite 
Kgment, whose sides stand upon the base of the first ; as BAD, 
which stands upon tlie segment BCD. (Fig. 25.) 

16. A tangent is a line touching a circle, which, produced, does 
sot cat it, as 6 A F. (Fig. 23.) 

IT. Circles are said to touch one another, which meet, but do not 
eat one another. 

18. Similar arcs, or similar sectors^ are those bounded by radii 
that make the same angle. 

19. Similar segments are those which contain similar triangles, 
ilike placed. 

20. A figure is said to be inscribed in a circle, or a circle circum- 
mhed about a figure^ when all the angular points of the figure are 
in tbe circumference of the circle. (Fig. 26.) 

21. A circle is said to be inscribed in a figure, or a figure circum^ 
Knhed about a circle, when the circle touches all the sides of the 
figure. (Fig. 27.) 

22. One figure is inscribed in awo/Aer, nvhen all the angles of the 
mscribed figure are in the sides of the other. (Fig. 28.) 

PROPOSITIONS. 

Phop. I. The radius C R, bisects any chord at right angles, which 
does not pass through the centre, as A B. (Fig. 29.) [Euc. B. III. 
Prop. 3.] 

Cor, 1. If a line bisects a chord at right angles, it passes through 
tie ccDtre of the circle. 

Cor. 2. The radius that bisects the chord also bisects the arc. 

II. In a circle equal chords are equally distant from the centre. 
[Edc. B. III. Prop. 14.] 

III. If 8e?eral lines be drawn through a circle, the greatest is the 
diaineter, and those that are nearest the centre are greater than those 
tbtt are farther off. fEuc. B. III. Prop. 15.] 

IV. If from any point three equal right lines can be drawn to the 
ojcamference, that point is the centre. [Euc. B. III. Prop. 9.] 

V. No circle can cut another in more than two points. [Euc. 
B. III. Prop. 10.] 

yi. There can be only two equal lines drawn from any exterior 
point, to the circumference of a circle. 

VIL In any circle, if several radii be drawn making equal angles, 
tie arcs and sectors comprehended thereby will be equal ; that is, if 
tbe angle A C B = B C D, then, the arc A B = B D, and the sector 
ACB = BCD. (Fig. 30.) [Euc. B. III. Prop. 26.] 

VIII. In the same or equal circles, the arcs, and also the sectors, 
ve proportional to the angles intercepted by the radii. 

IX. The circumferences of circles are to one another as their 
diuneters. 

X. A right line, perpendicular to the diameter of a circle, at the 



106 CIBCLBS, ETC. [PART I. 

extreme point, toncbes die circle in that point, and lies wholly withoot 
the circle ; or is a tangent to the circle. 

XI. If two circles- touch one another, either inwardly or outwardly, 
the line passing throogh their centres shall also pass through the 
point of contact. [Euc. B. III. Prop 11 and 12.] 

XII. In a circle the angle at the centre is double the angle at the 
circumference, standing upon the same arc; orBDC = 2BAC 
(Fi*r. 31.) Euc. B. III. Prop. 20.] 

XIII. All angles in the same segment of a circle are equal, as 
DAC = DBC, and DOC=DHC. (Fig 32.) [Eac. B. III. 
Prop. 21.; 

XIV. if two right lines DC, A B, be drawn from the extremities 
of two equal arcs D A, B C, they will be parallel. (Fig. 32.) 

XV. The angle A B C in a semicircle is a right angle. (Fig. 33.) 
[Euc. B. III. Prop. 31.; 

XVI. The angle FBO, in a greater segment FABCO, is less 
than a right angle ; and the angle D B E, in a less segment D B E, 
is greater than a right angle. (Fig. 33.) [Euc. B. III. Prop. 31.] 

X VII. If two lines cutting a circle, intersect one another in A ; 
and there be made at the centre, Z.ECF = Z.BAD; then the arc 
B D 4- G H = 2 E F, if A is within the circle ; or the arc B D — 
G H == 2 E F, if A is without. (Fig. 34.) 

XVIII. If from a point without, two lines, A B, AD be drawn 
to cut a circle ; the angle made by them is equal to the angle at the 
centre, standing on half the difference of the two arcs of the circum- 
ference G H, B D. (Fig. 31..) 

XIX. The angle A = Z.BHD-f-HDO, when A is witliin ; or 
A = B H D — H D G, when A is without the circle. (Fig. 34.) 

XX. In a circle, the angle made at the point of contact between 
the tangent and any chord, is equal to the angle in the alternate or 
opposite segment; ECF=EBC, and ECA=EGC. (Fig. 35.) 
[Euc. B. III. Prop. 32.] 

XXI. A tangent to the middle point of an arc, is parallel to the 
chord of it. 

XXII. If from any point B in a semicircle, a perpendicular B D 
be let fall upon the diameter, it will be a mean proportional between 
the segments of the diameter; that is, AD : DB :: DB : DC. 
(Fig. 30.) [Euc. B. VI. Prop. 13.] 

XXIII. The chord is a mean proportional between the adjoining 
segment and the diameter, from the similarity of the triangles : that 
is, A D : A B :: A B : AC ; and C D : CB :: C B : CA. (Fig. 36.) 

XXIV. In a circle, if the diameter A D be drawn, and from the 
ends of the chords A B, AC, perpendiculars be drawn upon the 
diameter ; the squares of the chords will be as the segments of the 
diameter ; that is, A E : A F :: A B' : AC". (Fig. 37.) 

XXV. If two circles touch one another in P, and the line PD£ 
be drawn through their centres ; and any line P A B is drawn 
through that point to cut the circles, that line 'will be divided in 
proportion to the diameters; that is PA : PB :: PD : PE. (Fig. 38.) 



CHlP.m.] CIRCLES, ETC. 107 

XXVI. If through any point F in the diameter of a circle, any 
chord, C F D be drawn, the rectangle of the segments of the chord is 
eqaal to the rectangle of the segments of the diameter ; C F x F D 
= AF X FB = also GF x FE. (Fig. 39.) [Euc. B. III. Prop. 35.] 

XXVII. If through any point F out of the circle in the diameter 
BA produced, any line F C D be drawn through the circle : the rect- 
ingle of the whole line and the external part is equal to the rectangle 
of the whole line passing through the centre, and the external part ; 
DFxFC = AF X FB = also F E X F G. (Fig. 40.) 

XXVIIL Let H F be a tangent at H ; then the rectangle C F x 
FD = square of the tangent F H. (Fig. 40.) [Euc. B. III. Prop. 36.] 

XXIX. If from the same point F, two tangents be drawn to the 
circle, they will be equal ; that is, F H = F I. (Fig. 40.) 

XXX. If a line P F C be drawn perpendicular to the diameter 
AD of a circle; and any line drawn from A to cut the circle and 
the perpendicular ; then the rectangle of the distances of the sections 
from Ay will ^be equal to the rectangle of the diameter and the 
distance of the perpendicular from A ; that is, A B x A C = 
AP X AD. (Fig. 41.) 

Also, A B X A C = A K\ (Fig. 41.) 

XXXI. In a circle E D F whose centre is C, and radius C E, if 
tbc points B, A, be so placed in the diameter produced, that C B, 
C£, CA be in continual proportion, then two lines BD, A D drawn 
from these points to any point in the circumference of the circle 
will always be in the given ratio of BE to A E. (Fig. 42.) 

XXXII. In a circle, if a perpendicular D B be let fall from any 
point D, upon the diameter C I, and the tangent D O drawn from D, 
tken AB, AC, AO, will be in continual proportion. (Fig. 43.) 

XXXIII. If a triangle B D F be inscribed in a circle, and a per- 
pendicular D P let fall from D on the opposite side B F, and the 
<iiMieter D A drawn ; then, as the perpendicular is to one side 
inclading the angle D, so is the other side to the diameter of the 
ciitle; that is, D P : D B :: D F : D A. (Fig. 44.) 

XXXI y. The rectangle of any two sides of an inscribed triangle 
is equal to the rectangle of the diameter, and the perpendicular on 
tl»e third side ; that is, BDxDF=ADxDP. (Fig. 44.) 

XXXV. If a triangle B A C be inscribed in a circle, and the angle 
A hiflectcd by the right line A E D, then as one side is to the seg- 
nicDt of the bisecting line within the triangle, so is the whole bisect- 
ing line to the other side; that is, A B : AE :: AD : A C; and 
ABxAC = BE.EC + AE2. (Fig. 45.) 

XXXVI. If a quadrilateral A BCD be inscribed in a circle, the 
mn of two opposite angles is equal to two right angles; that is, 
ADC-l-ABC= two right angles. (Fig. 46.) [Euc. B. III. 
Prop. 22.] 

XXXVII. If a quadrangle be inscribed in a circle, the rectangle 
of the diagonals is equal to the sum of the rectangles of the opposite 
mde»; or c A X BD = CB x DA + CD x A B. (Fig. 46.) 



108 CIRCLES, ETC. [pART I. 

XXXVIII. A circle is equal to a triangle whose base is the cir- 
cumference of the circle ; and height, its radius. 

XXXIX. The area of a circle is equal to the rectangle of half the 
circumference and half the diameter. 

XL. Circles (that is, their areas) are to one another as the squares 
of their diameters, or as the squares of the radii, or as the squares of 
the circumferences. [Euc. B. XII. Prop. 2.] 

XLI. Similar polygons inscribed in circles, are to one another as 
the circles wherein they are inscribed. 

XLII. A circle is to any circumscribed rectilineal figure, as the 
circle's periphery to the periphery of the figure. 

XLIII. If an equilateral triangle ABC be inscribed in a circle ; 
the square of the side thereof is equal to three times the square of the 
radius ; that is, A B* = 3 A D^. (Fig. 47.) 

XLIV. If from any point D in the circumference of a circle, 
having inscribed in it an equilateral triangle, chords be drawn to the 
three angles A, B, C ; the longest chord A D, is equal to the sum of 
the two lesser chords, B D and C D. (Fig. 48.) 

XLV. A square inscribed in a circle, is equal to twice the square 
of the radius. 

XLVI. The side of a regular hexagon inscribed in a circle, is 
equal to the radius of the circle. 

XLVII. If two chords in a circle mutually intersect at right 
angles, the sum of the squares of the segments of the chords is equal 
to the square of the diameter of the circle ; that is, A P^ -f P B* -r 
P C*^ -h P D' = diam.^. (Fig. 49.) 

XLVIII. If the diameter P Q be divided into two parts at any 
point R, and if R S be dmwn per]>endicular to P Q ; also R T ap- 
plied equal to the radius, and T R produced to the circumference 
at V: then, 

PR + RQ ., ., ., \ 

R T = ; or R T is the arithmehcal mean, ] between the 

f two segments 



RS = n/pR X RQ; or RS is the geometrical me&n. ;of the diame- 
« ^ « inT^ ' I ter P R, R Q 

RV = : or RV is the Aar mow iW mean. ^ ^^^f^' ^ 



PR + RQ 



XLIX. If the arcs P Q, QR, RS, &c., be equal, and there be 
drawn the chords P Q, PR, PS, &:c., then it will be P Q : P R :: 
PR : PQ -f PS :: PS : PR f PT :: PT : PS -f P V, &c. (Fig. 51.) 

L. If the arcs PQ, Q R, R S, &c. be equal, the angles Q P R, RP8, 
SPT,&c., will be equal ; or in equal circles, equal angles stand upon 
equal arcs, whether they be at the centres or circumferences. (Fig. 
51.) [Euc. B. III. Prop. 20.] 

LI. The centre of a circle being O, and P a point in the radius, 
or in the radius produced ; if the circumference be divided into as 
many equal parts A B, B C, C D, &c., as there are units in 2 n, and 
lines be drawn from P to all the points of division ; then shall the 
continual product of all the alternate lines, viz. PAx PC x PE, &c.. 



CHIP. III.] PLANES AND SOLIDS. 109 

be = r" — x" when P is within the circle, or = a? — r* when P is 
without the circle; and the product of the rest of the lines, viz., 
PB X P D X P F, &c., = r" -H ;i^ : where r = A O the radius, and 
« = 0P the distance of P from the centre. (Fig. 52.) 



Sect. VI. Of Planes and Solids. 

DEFINITIONS. 

1. The common section of two planes, is the line in which they 
meet, or cat each other. 

2. A line is perpendicular to a plane, when it is perpendicular to 
erenr line in that plane which meets it. 

3. One plane is perpendicular to another, when every line of the 
one, which is perpendicular to the line of their common section, is 
perpendicolar to the other. 

4. The inclination of one plane to another, or the angle they 
fonn between them, is the angle contained by two lines, drawn from 
anr point in the common section, and at right angles to the same, 
ODC of these lines in each plane. 

5. Parallel planes are such as being produced ever so far in any 
direction, will never meet, or which are everywhere at an equal per- 
pendicular distance. 

6. A solid an^ is that which is made by three or more plane 
tngles, meeting each other in the same point. 

7. Similar solids^ contained by plane figures, are such as have all 
their solid angles equal, each to each, and are bounded by the same 
nomber of similar planes, alike placed. 

8. A prism is a solid whose ends are parallel, equal, and like 
plane figures, and its sides, connecting those ends, are parallelograms. 
(Fig. 53 and 54.) 

9. A prism takes particular names according to the figure of its 
hue or ends, whether triangular, square, rectangular, pentagonal, 
hexagonal, &c. 

10. A right or upright prism^ is that which has the planes of the 
fldftf perpendicular to the planes of the ends or base. (Fig. 53.) 
When such is not the case it is called an oblique prism. (Fig. 54.) 

11. A parallelapipedj or parallelopipedon, is a prism bounded by 
Bx parallelograms, every opposite two of which are equal, alike, and 
parallel. (Fig. 55.) 

12. A rectangular parallelopipedon is that whose bounding planes 
>re all rectangles, which are perpendicular to each other. (Fig. 56,) 

13. A cube is a square prism, being bounded by six equal square 
sdea or faces, which are perpendicular to each other. (Fig. 57.) 

14. A cylinder is a round prism having circles for its ends ; and is 
conceived to be formed by the rotation of a right line about the 
cirtufflferences of two eqtuJ and parallel circles, always parallel to 
^axis. (Fig. 58.) 



110 SOLID GEOMETRY. [PART I. 

15. The axis of a cylinder is the right line A B joining the centres 
of the two parallel circles, about which the figure is described. 

IG. A pyramid is a solid whose base is any right-lined plane 
figure, and its sides triangles, having all their vertices meeting 
together in a point above the base, called the vertex of the pyramid. 
(Fig. 59.) 

1 7. Pyramids, like prisms, take particular names from the figure 
of their base. 

18. A cone is a round pyramid having a circular base, and is con- 
ceived to be generated by the rotation of a right line about the 
circumference of a circle, one end of which is fixed at a point aboYe 
the plane of that circle. (Fig. 60.) 

1 9. The axis of a cone is the right line, A B, joining the Tertex, 
or fixed point, and the centre of the circle about which the figure is 
described. 

20. When the axis of a cone or pyramid is perpendicular to the 
base, it is called a right cone or pyramid ; but if inclined it is called 
oblique, 

21. Similar cones and cylinders^ are such as have their altitudes 
and the diameters of their bases proportional. 

22. A sphere is a solid bounded by one curve surface, which is 
everywhere equally distant from a certain point within, called the 
centre. It is conceived to be generated by the rotation of a semi- 
circle about its diameter, which remains fixed. (Fig. 61.) 

23. The axis of a sphere is the right line about which the semi- 
circle revolves, and the centre is the same as that of the reyolving 
semicircle. 

24. The diameter of a sphere is any right line passing through 
the centre, and terminated both ways by the surface. 

25. The attitude of a solid is the perpendicular drawn from the 
vertex to the opposite side or base. 

PROPOSITIONS. 

Prop. I. If any prism be cut by a plane parallel to its base, the 
section will be equal and like to the base. 

II. If a cylinder be cut by a plane parallel to its base, the section 
will be a circle, equal to the base. 

III. All prisms and cylinders, of equal bases and altitudes, are 
equal to each other. [Euc. B. XI. Prop. 31.] 

IV. Rectangular parallclopipcdons, of equal altitudes, are to each 
other as their bases. [Euc. B. XI. Prop. 32.] 

V. Rectangular parallelopipedons, of equal bases, are to each 
other as their altitudes. 

VI. Because prisms and cylinders are as their altitudes, when 
their bases are equal : and, as their bases when their altitudes are 
equal. Therefore, universally, when neither are equal, they arc to 
one another as the product of their bases and altitudes : hence, also, 
these products are the proper numeral measures of their quantities or 
magnitudes. 



CHAP. III.] SOLID GEOMETRY. Ill 

VII. Similar prisms and cylinders are to each other as the cubes 
of their altitudes, or of any like linear dimensions. [Euc. B. XI. 
Prop. 33.] 

VIII. In any pyramid a section parallel to the base is similar to 
the base ; and these two planes are to each other as the squares of 
their distances from the vertex. 

IX. In a right cone, any section parallel to the base is a circle ; 
and this section is to the base as the squares of their distances from 
the vertex. 

X. All pyramids and cones of equal bases and altitudes, are equal 
to one another. 

XI. Every pyramid is the third part of a prism of the same base 
and altitude. [Euc. B. XII. Prop. 10.] 

XII. If a sphere be cut by a plane, the section will be a circle. 

XIII. Every sphere is two-thirds of its circumscribing cylinder. 

XIV. A cone, hemisphere, and cylinder of the same base and 
altitude, are to each other as the numbers 1, 2, 3. [Euc. B. XII. 
Prop. 10.] 

XV. All spheres are to each other as the cubes of their diameters; 
all these being like parts of their circumscribing cylinders. [Euc. 
B. XII. Prop. 18.] 

XVI. There are only three sorts of regular plane figures which 
can be joined together, so as to form a solid angle ; viz. three, four, 
or five triangles^ three squares^ and three pentagons: thus giving five 
different species of solid angles, which are those of the five regular 
or Platonic bodies ; viz. : — 

1. The tetraedron^ (^^g* ^2,) bounded by four equilateral tri- 
angles^ each solid angle of which is formed by three triangles, 

2. The kexaedron^ or cuhcy (Fig. 57,) contained by six squares^ 
each solid angle of which is formed by three squares. 

3. The octaedron, (Fig. 63,) bounded by eight triangles^ each 
solid angle of which is formed by four triangles, 

4. The dodecaedron^ (Fig. 64,) bounded by twelve pentagons^ 
each solid angle of which is formed by three pentagons, 

5. The icosaedron, (Fig. 65,) bounded by twenty triangles, each 
solid angle of which is formed by five triangles, 

XVII. Only one sort of the foregoing five regular bodies, joined 
at their angles, can completely fill a solid space, without leaving any 
Tacuity ; viz. eight hexaedrons, or cubes. 

XVIII. A sphere is to any circumscribing solid B F, (all whose 
planes touch the sphere,) as the surface of the sphere to the surface 
of the solid, r Fig. 66.) 

XIX. All bo£es cii*cumscribing the same sphere, are to one an- 
other as their surfaces. 

XX. The ^here is the greatest or most capacious of all bodies of 
equal surface. 



112 PRACTICAL GEOMETRY. [PART I. 

Sect. VII. Practical Geometry. 

It is not intended in this place to present a complete collection of 
Geometrical Problems, but merely a selection of the most useful, 
especially in reference to the employments of Mechanics and En- 
gineers. 

The instruments for the purposes of geometrical construction, are 
too well known to require any description here; and their use is 
much easier learned by an examination of the instruments themselves, 
than by any written explanation. 

Prob. I. From a given point B in a given straight line A C, to 
draw a line perpendicular to the same. 

From B as a centre, witli any radius, describe arcs cutting A C in 
a and c, then from a and c with any larger radius describe arcs 
cutting each other in d and e ; then the straight line d'Be vnW be 
perpendicular to A C. (Fig. 6'7.) 

Prob. II. To erect a perpendicular at the end of a given line. 

With any radius, and from a point somewhere above A B, describe 
a semicircular arc passing through the point B, at which the perpen- 
dicular is to be erected, and also cutting A B in C ; then, through C 
and the centre of the arc produce the right line C D, cutting the arc 
in D, and a line joining D and B will be the perpendicular required. 
(Fig. 68.) 

Prob. III. To bisect any given angle A C B. 

From C as a centre, with any radius, describe an arc cutting the 
sides in D and £ ; then from D and £, as centres with the same 
radius, describe arcs cutting each other in F ; then the straight line 
joining C and F will bisect the angle A C B. (Fig. 69.) 

Prob. IV. To bisect a given angle BAP; then to bisect its half; 
and so on. 

Through any point B draw B E parallel to A P, and upon B E set 
off the distance B C equal to B A ; then join A C^ and it will bisect 
the angle BAP. 

Again, set off, upon B E, from C, C D = C A ; join A D, and it 
will bisect C A P, or quadrisect BAP. 

Again, set off, upon B £, D E = D A ; join £ A ; and E A P will 
be ^ of B A P : and so on. (Fig. 70.) 

Prob. Y. At a given point A in a given line A B, to make an angle 
equal to a given angle C. 

From the centres A and C, with the same radius, describe the arcs 
D E, F G, Then, with radius D £, and centre F, describe an arc, cut- 
ting F G in O. Through G draw the line A G ; and it will form the 
angle required, (Fig. 71 .) 

Prob. VI. To divide any given angle ABC into three equal parte. 

From B, with any radius, describe the circle A C D A. Bisect the 
angle A B C by B £, and produce A B to D. On the edge of a ruler 



CHAP. III.] PRACTICAL GEOMETRY. 113 

mark off the length of the radius A B. Lay the ruler on D, and 
move it till one of the marks on the edge intersects B £, and the 
other the arc A C in O. Set off the distance C G from G to F : and 
draw the lines B F, and B O, they will trisect the angle ABC. 
(Fig. 72.) 

Prob. VII. To divide a given line A B into any proposed number 
of equal parU, 

Ist Method. Draw any other line AC, forming any angle with the 
given line A B ; on which set off as many of any equal parts, A D, 
D E, E F, F C, as the line A B is to he divided into. Join B C ; 
parallel to which draw the other lines F G, EH, D I : then these 
will divide AB in the manner required. (Fig. 73.) 

2nd Method^ without drawing parallel lines. Let A B he the line 
which is to be divided into n equal parts. Through one extremity 
A draw any right line A D, upon which set off n -f 1 equal parts, the 
point D being at the termination of the (n + l)th part. Join DB 
and produce it until the prolongation BE = B D. Let F be the 
termination of the (n — l)th part. Join FE, and the right line of 
junction will cut the given line AB in the point P, such that 

B P = - A B ; and of course n distances each equal to B P set off 

upon B A, will divide it, as required*. (Fig. 74.) 

Pbob. VIII. To cut off from a given line A B, supposed to he 
tery shorty any proportional part. 

Suppose, for example, it were required to find the -y^, -^^^ j\r, &c. 
of the line A B, fig. 75. From the ends A and B draw AD, B C, 
perpendicular to A B, and divide A D into twelve equal parts ; then 
through these divisions 1, 2, 3, &c., draw lines \f 2^, &c., parallel 
to A B. Draw the diagonal A C, and 1 d will be the -^-^ of A B ; 
2 c, -^j, and so on. The same method is applicable to any other 
part of a given line. 

Prob. IX. To make a diagonal scale^ say^ offeety inches, and 
tenths of an inch. 

Draw an indefinite line A B, on which set off from A to B the 
given length for one foot, any required number of times ; and from 
these divisions A, C, H, B, draw AD, CE, &c., perpendicular to AB. 
On A D and B F set off any length ten times, and through these 
divisions draw lines parallel to A B ; then divide A C and D E into 
twelve equal parts, each of which will be one inch. Draw the lines 
A 1, ^2, &c., and they will form the scale required; viz., each of 
the larger divisions from E to G, G to F, &c., will represent a foot ; 
each of the twelve divisions between D and £, an inch ; and the 

* The truth of thie method is easily demonstrated. Through i the inter- 
mediate point of division, on a n, between f and d, draw i b. Then, because 
D B « B K, and D I B 1 F, I B is parallel to F P. Consequently, b P : B a : : 
I P : I A : : 1 : n, by construction. 

I 



114 PRACTICAL OBOMBTBY. (^r^^. . 

several horizontal lines parallel to R C in the triangle £ C R, will be 
equal to -j\^, -f%, -,»ff, &c., of an inch. (Fig. 76.) 

Note. — If the scale be meant to represent feet, or any other onit, 
and tenths and hiindredfks^ then D E must be divided into ten instead 
of twelve equal parts. 

Prob. X. To fitid the centre of a circle. 

Draw any chord A B, and bisect it perpendicularly with the line 
R D. Then bisect R D in C, which will be the centre required. 
(Fig. 29.) 

Prob. XI. To divide a given cirde into any number ofeoneentrk 
parts^ equal to each other in area. 

Draw the radius A B, and on it describe the semicircle Afe dB. 
Divide AB into the proposed number of equal parts, 1, 2^ 3, &c., 
and erect the perpendiculars 1 rf, 2e, Sf &c., meeting the semicircle 
in rf, e,yj &c. Then from the centre B, and ^nth radii Be/, Be, &c. 
describe circles ; so shall the given circle be divided into the proposed 
number of equal concentric parts. (Fig. 77.) 

Prob. XII. To divide a given circle into any number qf partij 
equal both in area and perimeter. 

Divide the diameter Q R into the proposed number of equal parts 
at the points S, T, V, &c. ; then, on one side of the diameter 
describe semicircles on the diameters QS, QT, QV, and on the 
other side of it describe semicircles on RV, RT, RS; so shall the 
parts 17, 35, 53, 71, be all equal> both in area and perimeter. 
(Fig. 78.) 

Prob. XIII. To describe the circumference of a circle through three 
given points^ A, B, C. 

From the middle point B draw chords B A, B C, to the two other 
points, and bisect these chords perpendicularly by lines meeting in O, 
which will be the centre. Then from the centre O, at the distance 
of any of the points, as O A, describe a circle, and it will pass 
through the two other points B, C, as required. (Fig. 79.) 

The same method may be employed for finding the centre of 
a circular arc, by taking any three points in the same, as A, B, 
and C. 

Prob. XIV. To describe mechanically the circumference of a cireU 
through three given points^ A, B,C, tchen the centre is inaccessible^ or the 
circle too large to be described with compasses. 

Place two rulers M N, R S, cross ways, touching the three points 
ABC. Fix them in V by a pin, and by a tmasverse piece T. Hold 
a pencil in A, and describe the arc B A C, by moving the angle RAN 
so as to keep the outside edges of the rulers against the pins B C. 
Remove the instrument R V N, and on the arc described mark two 



CHAP. III.] PRACTICAL GEOMSTBY. lid 

points, D, E, 80 that their distance Bhall be equal to the length B C. 
Apply the edges of the instrument against D £, and with a pencil in 
6 describe the arc B C, which will complete the circumference of the 
circle required. (Fig. 80.) 

Otherwise, — Let an axle of 12 or 15 inches long carry two unequal 
wheels A and B, of which one, A, shall be fixed, while the other, B, 
shall be susceptible of motion along the axle, and being placed at any 
assigned distance, A B, upon the paper or plane on which the circle is 
to be described. Then will A and B be analogous to the ends of a 
conic frustum, the vertex of the complete cone being the centre (O) 
of the circle (C D E F) which will be described by the rim, or edge, of 
the wheel A, as it rolls upon the proposed plane. Then it will be, 
as the diameter of the wheel A is to the difference of the diameters 
of A and B, so is the radius of the circle proposed to be described by 
A, to the distance, A B, at which the two wheels must be asunder, 
measured upon the plane on which the circle is to be described. 

The wheel B will evidently describe, simultaneously, another circle 
(O H I K) whose radius will be less than that of the former bv A B. 
(Fig. 81.) 

Pbob. XV. On a ^ven chord ABto describe mechanicaUy an arc 
of a large circle that shall contain any number of degrees. 

Place two rulers, forming an angle A C B, equal to the supplement 
of half the given number of degrees, and fix them in C* Place two 
pins at the extremities of the given chord, and hold a pencil in C ; 
then move the edges of this instrument against the pins, and the 
pencil will describe the arc required. (Fig. 82.) 

Suppose it is required to describe an arc of 50 degrees on the given 
chord A B ; subtract 25 degrees (which is half the given angle) from 
180, and the difference, 155 degrees, will be the supplement. Then 
form an angle A C B of 155 degrees with the two rulers, and proceed 
«s has been shown above. 

Pbob. XVI. To describe the segment of a circle of large radius^ of 
which the chord A B and versine C D are given. 

Through D draw H I parallel to A B ; then join A D and D B, and 
draw A H perpendicular to A D and B I perpendicular to B D, also 
draw A d and B d perpendicular to A B. Then divide A C, B C, H D, 
and I D, into any number of equal parts, 1, 2, 8, &c., and draw lines 
joining the corresponding numoers in H I and A B ; also divide A c/, 
B d, into the same number of equal parts, a, by c^ &c., and draw lines 
from the point D to these last divisions, then will the points where 
the lines a D, 6 D, c D, &c. cut the lines 1 1, 22, 33, &c., be so many 
points in the required segment. (Fig. 83.) 

Pbob. XVII. To find the length of any given arc of a circle^ A B. 

From A and B, as centres, widi radius equal to a quarter of the 
diord of the arc A B, describe arcs cutting the given arc in C, and its 
diord ID D ; join C D, which will equal h^f the length of the aro A B 
nearly. (Fig. 84.) 

I 2 



116 PRACTICAL GEOMETRY. [PART I. 

Prob. XVIII. To draw a straight line equal in length to any given 
portion of the circumference of a circle. 

Let A B C D be the circle, and let A a, a ^, 6 B, be the portions of 
the circumference of wliich the length is required ; draw^ the two 
diameters B D and A C at right angles to each other, and at the end 
of the latter draw the tangent line Ae ; then divide the radius EC 
into four equal parts, and set off three of them from C to F on the 
diameter produced ; then draw lines from F through the points a,d,B, 
cutting the tangent line A^, in c, rf, e; so shall the lines Ac, c</, de, 
be equal in length to the arcs A a, a 6, 3 B, and the whole line Ae 
equal to the quadrant A B. (Fig. 85.) 

Prob. XIX. To bisect any given triangle ABC. 

Upon any one of the sides, as A B, describe a semicircle, which 
bisect in the point D ; then from B as a centre with radius B D de- 
scribe an arc cutting A B in E, through which point draw the line 
E F parallel with the side A C ; then will the line £ F bisect the 
given triangle ABC. (Fig. 86.) 

Prob. XX. To reduce a given rectilinear figure, ABCDEFOA, 
of any number of aides, to a triangle of equal area. 

Join any two alternate angles, as A C, and through the interme- 
diate angle B, draw B H parallel to A C, cutting one of the adjoining 
sides in H, and join C H, then will the triangle C B a, added to the 
figure, be equal to the triangle a A H taken out of the same ; in like 
manner join H D, and draw G I parallel to the same through the in- 
termediate point O, producing it to cut the side A G (also produced) 
in I, and join I D. Next join D F, and through E draw a line paral- 
lel thereto, cutting the side G F produced in K, and join D K ; lastly, 
join D G, and draw parallel to the same through the point K a line 
cutting the side A G produced in L, then join D L, and the tri- 
angle I D L will be equal in area to the given rectilinear figure 
ABCDEFGA. (Fig. 87.) 

Prob. XXI. To form a rectangle of the largest area, in a given 
triangle, ABC. 

Bisect any two sides A B and B C in D and E, and from those 
points draw lines D F and £ G perpendicular to the third side A C, 
and join D E, then will the rectanele D E F G be the largest which 
can be inscribed in the given triangle. (Fig. 88.) 

Prob. XXII. To form a square equal in area to a given triangley 
ABC. 

On the longest side produced set off C D, equal half the perpendi- 
cular height B E ; and at C erect the perpendicular C F ; then on 
A D describe a semicircle cutting C F in G, and on Q G form the 
square CGHI, which will be equal in area to the given triangle. 
(Fig. 89.) 



CHAP. III.] PRACTICAL OEOIIBTRY. 117 

Pbob. XXIII. To find the 9ide of a square eqtuU in area to a given 
rectangle^ A B C D. 

Produce the lesser side A B of the rectangle till A E equal the 
longer side AD; then describe upon A £ a semicircle cutting B C in 
F, and join A F, which will be the side of the square required. 
(Fig. 90.) 

Pbob. XXIV. To find the side of a square egtmlto the difference 
beiveen two given squares. 

Let the sides of the two given squares be A and B. Then draw 
the line C D equal to the lesser line B, and at one of its extremities 
erect the indefinite perpendicular D £ ; then with radius equal 
to A, and from C as a centre, describe an arc cutting this perpen- 
dicular in E ; then D E will equal the side of the square required. 
(Fig. 91.) 

Pbob. XXV. To find the side of a square equal in area to any 
number of given squares. 

Let the lines A, B and C be the sides of the given squares. Draw 
DE equal to A, and at the end of it erect the perpendicular E F equal 
to B ; join D F and perpendicular to it, from the point F, erect the 
perpendicular O F equal to C ; then join D O, which will be equal 
to the side of a square equal in area to the three squares on A, B, 
and C. (Fig. 92.) 

Pbob. XXVI. To find the side of a square nearly equal in area to 
a given circle, A B C D. 

Draw the two diameters, AC, B D, at right angles to each other ; 
then bisect the radius E C in F, and through D and F produce the 
straight line D G cutting the circle in G, then will D G equal the side 
of the square required. (Fig. 93.) 

Pbob. XXVII. Given the side of a regular polygon of any number 
of tides (not exceeding twelve% to find the radius of the circle in which 
it may be inscribed. 

Multiply the given side of the polygon by the number which stands 
in column 6, opposite its proper name in the annexed Table of 
Polygons ; the product will be the radius required. 

Thus, suppose the polygon was to be an octagon, and each side 1 2, 
then 1-3065628 X 12 = 15-6687536 would be the radius sought. 
Take 15*67 as a radius from a diagonal scale, describe a circle, and 
from the same scale, taking off 12, it may be applied as the side of 
an octagon in that circle. 

Pbob. XXVIII. Given the radius of a circle, to find the side of any 
regular polygon (not having more than twelve sides) inscribed in it. 

Multiply the given radius by the number in column c, standing 
opposite the number of sides of the proposed polygon : the product 
is the length of the side required. 

Thus, suppose the radius of the circle to be 5, then 5 x 1*732051 
as 8*66025, will be the side of the inscribed equilateral triangle. 



118 



PKAOTICAL OSOMITBT. 

TABLE OP POLYGONa 



[Fi 



6 "5 




Multipliers 


Radius of 


Fac 


Names. 


for areas. 


circmD. cir. 


fori 


Z,'^ 




(«) 


(ft) 


( 


8 


Trigon 


0-4330127 


0-5773503 


1-73S 


4 


Tetragon, or Square 


1-0000000 


0-7071068 


1-41^ 


5 


Pentagon 


1-7204774 


0-8506508 


117^ 


6 


Hexagon 


2-5980762 


10000000 


1-00( 


7 


Heptagon 


3-6339124' 


1- 1523824 


0-86' 


8 


Octagon 


4-8284271 


1-3065628 


0-76, 


9 


Nonagon 


6-1818242 


1-4619022 


0-68- 


10 


Decagon 


7-6942088 


1-6180340 


0-61J 


11 


Undecagon 


9-8656399 


1-7747324 


0-56; 


12 


Dodecagon 


111961524 


1-9318517 


0-51' 



Pros. XXIX. To reduce a simple rectilinear figure to a i 
one upon either a smaller or a larger scale. 

Pitdi upon a point P any where about the given figure A B 
either within it, or without it, or in one side or angle; but m 
middle is best. From that point P draw lines through all the i 
upon one of which take P a to P A in the proposed proportion 
scales, or linear dimensions ; then draw a b parallel to A B, 
B C, &c. ; so shall abode be the reduced figure sought, 
greater or smaller than the original. (Hutfons Mens.) (Fig. S 

Otherwise to Reduce a Figure by a Scale, — Measure all the 
and diagonals of the figure, as A B C D E, by a scale ; and lay 
the same measures respectively from another scale, in the pro] 
required. 

To Reduce a Map^ Design^ or Figure^ by Squares. — Divide tl 
ginal into a number of little squares, and divide a fresh paper, 
dimensions required, into the same number of other squares, 
greater or smaller, in the proportion required. Then, in every 
of the second figure, draw what is found in the corresponding 
of the first or original figure. 

The cross lines forming these squares may be drawn with a 
and rubbed out again after the work is finished. But a more 
and convenient way, especially when such reductions are 
wanted, would be to keep always at hand frames of squares 
made, of several sizes ; for, by only just laying them down up 
papers, the corresponding parts may be readily copied. These : 
may be made of four stifiT or inflexible bars, strung across with 
hairs, or fine catgut. 

When figures are rather complex, the reduction to a differen 
will be best accomplished by means of such an instrument as I 
sor Wallace's Eidograph^ or by means of a Pantograph^ an i 
ment which is now considerably improved by simply changic 
place of the fulcrum. See the Mechanics' Orade^ Part II. p. ; 



CfliP.IV.J WS10HT8 AND IfEASURKS. 119 



CHAP. IV. 
MENSURATION. 

Sect. I. Weights and Measures, 

MmuEATiON is the application of arithmetic to geometry, by 
wliicb m ire enabled to discover the magnitude and dimeneious of 
tnj geometrical figures, whether solid or superficial. To enable us 
to expitis this magnitude in determinate terms, it is necessary to 
womeflome magnitude of the same kind as the unit, and tlien, by 
itatiDg how many times the given magnitude contains that unit, we 
obtain its measure. 

The different species of magnitude which have most frequently to 
be determined, are distinguishable into six kinds, viz. : — 

1. Length. 

2. Surface. 

3. Solidity, or Capacity. 

*. force of Gravity, commonly called Weight. 

5. Angles. 

6. Time. 

The several units assumed as the standards of measurement of each 
of then particular species of magnitude, are entirely arbitrary, and, 
ttoie^ently, vary among different nations. In this kingdom they 
w heea fixed by Act of Parliament*, and are as follows, viz. : — 

Length is a yard. 

. Surface is a square yard, the ^g^^^ th of an acre. 
I (Solidity is a cubic yard. 

Tie standard of < i^^P???^ '' "" ^^l'"''' 
mmuuMuix "» \ height IS a pound. 

I Angular measurement is a degree, the d60th part 

of the circumference of the circle. 

Time is a day. 

ne values of the whole of the above are determined directly or 
Wifectly by comparison with the length of a pendulum, which in 
^ latitude of London, placed at the level of the sea, and in a 

* 5 Gso. IV. c. 74 ; which took effect Itt January, 1826. 



120 WBIOHTS AND 1IBA8URBS. [r^^. 

yacuuTD, would vibrate seconds of mean time. The length of such 
a pendulum being, to the length of the standard yard, as 39*1393 
inches are to 36. 

Since the passing of this act, however, some very elaborate and 
scientific experimenis of Mr. Francis Daily have shown that errors 
of sufficient moment to be taken into the account in an inquiry of 
this kind, render the above proportion inaccurate*. We do not, in 
fact, yet know the length of a seconds* pendulum at London, vibrat- 
ing in the circumstances proposed. 

The following standard yards, made with great accuracy, give the 
annexed results : — 

Inches. 
General Lambton's scale, used in India ... 35*99934 

Sir George Shuckburgh's scale 35*99998 

General Ray's scale 36*00088 

Royal Society's standard 3600135 

Ramsden'sbar 36002*9 

Its copy, at Marischal College, Aberdeen... 36*00244 

1. MEASURES OF LENGTH. 
Inchcft. Feet. 

12 = 1 Yard.. 

o/» o __ I Rod* 

<5t> — d — 1 orPoie*. 

198 = 16J = 5J == 1 Furicmg.. 

7920 = 660 = 220 = 40 = 1 Mile. 
63360 = 5280 = 1760 = 320 = 8 = 1 

The mean length of a degree of latitude measured on the terrestrial 
meridian, is 69*0444 imperial miles; the 60th part of which, or 
6075*6 feet, is the length of a nautical or geographical mile, three of 
which are equal to a league. And the length of a degree of lo/igi- 
tude, measured upon the equator, is 69*1555 imperial miles. 

An inch is the smallest lineal measure to which a name is nven ; 
but subdivisions are used for many purposes. A mong mechanics the 
inch is commonly divided into eighths. By the officers of the revenue, 
and by scientific persons, it is divided into tenths.^ hundredthn^ &c. 
Formerly it was made to consist of 12 parts, called lines^ but these 
have properly fallen into disuse. 

Particular Measures of Length. 
A Nail = 2\ Inches \ 

Y A — • 4 O f i ^^^ ^^^ measuring cloth of all kinds 

Ell = 5 Quarters ) 

Hand = 4 Inches Used for the height of horses. 

Fathom = 6 Feet Used in measuring depths. 

Link = 7 Inc., 92 \ Used in Land Measure, to faciliti 

hdths. } computation of the content, 
Chain =100 Links ) square chains being equal to an ac 

* See foot-note, p. 223. 



f 



CRIP. IV.J WBIOHTS AND MKA8UBES. 121 

2. MEASURES OF SURFACES. 



SqoRliKfaa. 

144 = 

1296 = 

39204 = 

1568160 = 


Sq.Feet. 

1 

9 = 

272i = 

10890 = 


Sq. Yards, 

1 orRodi. 

30| = 1 Rood.. 

J210 = 40= 1 a™. 


6272640 = 


43560 = 


4840 = 160= 4= 1 mL 


4014489600 = S 


^878400 = 


3097600 =102400 = 2560 = 640 = 1 



d. HBASUBE8 OF SOLIDITY AND CAPACITY. 

Division I. Measures 0/ Solidity, 

Cubic Inchet. Cubic Feet. 

1728 = 1 Cubic Yard. 

46656 = 27 = 1 

Division II, 

Mntures of Capacity for aU liquids^ and for aU dry goods^ except 
wc^oiare comprised in the third division, 

Tbe imperial gallon (the standard for all measures of capacity) 
contains 10 pounds imperial avoirdupois weight of distilled water, 
'wghedin air at 62^ Fahr. (the barometer being at 30 inches) ; con- 
'wiuently, its capacity is 277*274 imperial cubic inches. 

4 Gills = 1 Pint = 34-659 Cubic Inches. 

2Rnt8 = 1 Quart = 69 318 „ „ 

4 Quarts = 1 Gallon = 277*274 „ „ 

2 Gall. =1 Peck = 554548 „ „ 

8 Gall. = 1 Bushel = 2218-192 „ „ 

8 Bush. = 1 Quarter = 10-269 Cubic Feet. 

5Qr8. = 1 Load = 51*345 „ „ 

The four last denominations are used for dry goods only. For 
liquids leveral denominations have been heretofore adopted, viz. : — 
^or beer, the firkin of 9 gallons, the kilderkin of 18, the barrel 
0(36, the hogshead of 54, and the butt of 108 gallons. These 
will probably con tin ae to be used in practice. For wine and 
"pvits, there are, the anker, runlet, tierce, hogshead, puncheon, 
PpC) butt, and tun; but these may be considered rather as the 
i^Anies of the casks in which such commodities are imported, than 
•• expressing any definite number of gallons. It is the practice to 
8>oge all such vessels, and to charge them according to their actual 
cwitcnt. 

Flour is sold, nominally, by measure, but actually by weight, 
'^Itoned at 7 lb. aToirdupois to a gallon. 



122 



WBI0HT8 AND IfEASURIS. 



[PABT I, 



Division HI. 

Imperial Measures of Capacity for coalsy 'culm^ lime, fish^ potaioet^ 
fruity and other goods, commonly sold by hbapbd measukb : 



2 Gallons = 1 Peck 

8 Oallons = 1 Bushel 

3 Bushels = 1 Sock 
12 Sacks == 1 Chald. 



28154 I ^^^^^ Inches, nearly, 
-gt I Cubic Feet, nearly. 



The goods are to be heaped up in the form of a cone, to a height 
above the rim of the measure of at least f of its depth. The out- 
side diameter of measures used for heaped goods are to be at least 
double the depth ; consequently not less than the following dimen- 



19i inches. 



Bushel 

Half-bushel 15| inches. 



! 



Peck 12 J inches. 
Gallon 9| inches. 
Half-gallon 7f Inches. 



The imperial measures described in the second and third divisiont 
were established by Act 5 Geo. IV. c. 74. Before that time there 
were four different measures of capacity used in England, yiz. : — 

2. For malt liquors, the gallon of which con- ) ooo 

tained j ^^^ " 

3. For com and all other dry goods, not | aoo.o 

heaped, the gallon of which contained ... ) " " 

4. For coals, which did not differ sensibly from the imperial 

measure. 

Hence, with respect to wine, ale, and com, it will be expedient 
to possess a 

Table of Factors, 
For converting old measures into new, and the contrary. 





By Decimals. 


By Vulgar Fractions 
nearly. 


Com 
Measure. 


Wine 
Measure. 


Ale 
Measure. 


Corn 
Mea- 

sure. 


Wine 
Mea- 

sure. 


Ale 
Mea- 

sure. 


To convert old ) .96943 
measures to new. S 


-83311 


1-01704 


n- 


f 


n 


To convert new J ^ 3^ 3 
measures to old. S 


1-20032 


-98324 


i\ 


i 


^ 



N.B. For reducing the prices, these numbers must all be revenad. 



GHAP. IV.] WUOHTS AND IfSAtUBBS. 123 

4. MBA8URE8 OF WEIGHT. 

Division L Avoirdupois Weight. 

The standard of weight is the avoirdupois pound, the value of 
'vhich is determined by its heing the weight of 27'7274 cuhic inches 
^f distilled water, weighed in air at 62"^ Fahr., with the harometer at 
SO inches. 

Avoirdupois weight is used in almost all commercial transactions, 
^md in the common dealings of life; its divisions are as follows : — 



Dndum. 




Ounees. 














16 


s: 


1 




Pounds. 










256 


S5 


16 


=c 


1 


QiuDten. 






7168 


= 


448 


=s 


28 




1 


Hundred- 
weighu. 




28672 


= 


1792 


= 


112 


= 


4 


5= 1 


Ton. 


573440 


z^ 


35840 


= 


2240 


= 


80 


= 20 






Particular weighu belonging to this Division. 












cwt. qr. 


lb. 








8 Pounds = 


1 Stone 


z= 





8 


Used for meat. 




1* .> 


= 


1 « 


s= 





14 \ 








2 Stone 


s= 


1 Tod 


s= 


1 











e|Tod 


= 


1 Wev 


= 


1 2 


14 

I) 


Used 


in the wool trade. 


2 Wejs 


=s 


1 Sack 


== 


3 1 








12 Sacks 


2= 


1 Last 


=5 


30 









Division IL Troy Weight, 
For weighing gold, silver, and precious stones (except diamonds), 
troj weight is employed, its divisions are — 

Penny- 
OtaiBi (gn.) i*«ighU (dwts.) 



24 


=s 


1 


OUBCM(OI.) 


480 


5= 


20 


= 1 Pound (lb.) 


5760 


= 


240 


= 12 = 1 



But troy weight is also used hv apothecaries in compounding medi- 
cines; and 18 then divided as follows, viz. : — 



Onant(gr.) Seruplo (3.) 

20 = 1 Drachms 


(3.) 






60 

480 

5760 


i 


3 = 
24 = 

288 ^ 


1 

8 

96 




Ounces (3 ) 
= 1 
= 12 


Pound (lb.) 
= 1 



The troy pound is equal to the weight of 22*8157 cuhic inches of 
distilled water, weighed in air at 62^ Fahr., and the harometer at 
30 inches; and is, therefore, less than the avoirdupois pound, in the 
proportion of 144 to 175. 



124 WEIGHTS AND MEASURES. [PABT I. 

OS. dwts. gn. 

1 lb. avoirdupois = 14 11 15i troy = 7000 troy grains. 
1 oz. „ == 18 5$ „ = 437-5 „ 

1 dr. „ =01 3| „ = 27-343 „ 

1 trov lb. = 0-822857 avoirdupois lb. 

1 avoir, lb. = 1-215271 troy lb. 

For scientific purposes the grain only is used ; and sets of weights 
are constructed in decimal progression, from 10,000 grs. downwards 
to i^J^th of a grain. 

The carat^ used for weighing diamonds, is 3^ grains. The term, 
however, when used to express the fineness of gold, has a relative 
meaning only. Every mass of alloyed gold is supposed to be divided 
into 24 equal parts; thus the standard for coin is 22 carats fine, that 
is, it consists of 22 parts of pure gold, and 2 parts of alloy. What 
is called the new standard^ used for watch-cases, &c. is 18 carats 
fine. 

5. Angular Measure; or. Divisions of the Circle. 
60 Seconds " =1 Minute, denoted by ' 

60 Minutes = 1 Degree, „ ° 

30 Degrees = 1 Sign „ ' 

90 Degrees = 1 Quadrant. 

360 Degrees, or 12 Signs = 1 Circumference. 

Formerly the subdivisions were carried on by sixties ; thus, the se- 
cond was divided into 60 thirds, the third into 60 fourths, &c. At 
present, the second is more generally divided decimally into lOths, 
lOOths, &c. The degree is frequently so divided. 

6. Measure of Time. 

60 Seconds = 1 Minute. 

60 Minutes = 1 Hour. 

24 Hours = 1 Mean Solar Day. 

23 H. 56 M. 3-5 S. = 1 Sidereal Day. 

7 Days = 1 Week. 

28 Days = 1 Lunar Month. 

28, 29, 30, or 31 Days = 1 Calendar Month. 

12 Calendar Months = 1 Year. 

365 Days = 1 Common Year. 

366 Days = 1 Leap Year. 
365jDavs = 1 Julian Year. 
365 D. 5 H. 48 M. 45| S. = 1 Solar Year. 
365 D. 6 H. 9 M. 11 S. =1 Sidereal Year. 

A solar day is the time that elapses between two successive transits 
of the sun over the same meridian, and is not always of equal dura- 
tion, being longer at some seasons of the year than at others ; the 
difference between the actual length of a solar day and the mean 
length, is called the equation of time. A sidereal day is the interval 
of time that elapses between two successive transits of any fixed star 



CHiP, IV.] 



WEIGHTS AND MRA8UBBS. 



125 



OTer the same meridian, and is tbe most uniform of all astronomical 
pen'ods, neither theory nor observation having detected the slightest 
TSjiatioD in its length. A solar year is the time in which the earth 
passes through the twelve signs of the zodiac, and is the natural year, 
Dccause it always keeps the same seasons in the same months. The 
wd^eal year is the time that elapses between the eartb leaving any 
fised star and returning to it again. 

In 400 years, 97 are leap years, and 303 common. 

Tbe same remark, as in the case of angular measure, applies to tbe 
mode of subdividing the second of time. 

COMPARISON OF ENGLISH AND FRENCH WEIGHTS AND MEASURES. 

The following is a comparative Table of the Weights and Measures 
^i England and France, which were published by tbe Royal and 
dJoitral Society of Agriculture of Paris, in the Annuary for 1829, 
^uid founded on a Report, made by Mr. Mathieu, to the Royal Aca- 
demy of Sciences of France, on the bill passed the 17th of May, 1824, 
^■^latiTe to the Weights and Measures termed " Imperial," which are 
xaow used m Ghreat Britain. 



ENGLISH. 

1 Inch (l-36th of a yard) 
1 Foot (l-3rd of a yard 
'Yard imperial 
Fathom (2 yards) . 
^ole, or perch (5^ yards 

Fnrlong (220 yards) 

Mile (1760 yards) 

FRENCH. 



1 Mill; 

1 Centimetre 

1 Decimetre . 

IMetre . 

^yriamctre , 



Measures of Length. 

FRENCH. 

2*539954 centimetres. 
3-0479449 decimetres. 
0-91438348 metre. 
1-82876696 metre, 
5-02911 metres. 
201-16437 metres. 
1609*3149 metres. 

ENGLISH. 

0-03937 inch. 
0-393708 inch. 
3-937079 inches. 
39-37079 incbes. 
3-2808992 feet. 
1-093633 yard. 
6*2138 miles. 



Square Measures. 



ENGLISH. 



J Yard sqoare 
Jod (square perch) 
1 Bood (1210 yards square) 
' Acre (4840 yards square) 

FRENCH. 

J Metre square 

\^ 

IHeetare . . 



FRENCH. 



0-836097 metre square. 
25*291939 metres square. 
10-116775 ares. 

0-404671 hectare. 

ENGLISH. 

1-196033 yard square. 
098845 rood. 
2-473614 acres. 



126 



WnOHTS AND ICEABOBKt. 



[pAWr I 



Solid Measures, 



ENGLISH. 

1 Pint (l-8th of a gallon) 
1 Quart (l-4th of a gallon) 
1 Gallon imperial . 
1 Peck (2 gallons) 
1 Bushel (8 gallons) 
1 Sack (3 bushels) 
1 Quarter (8 bushels) 
1 Chaldron (12 sacks) 

FRENCH. 



1 Litre . . = . 

I Decalitre . . . . = . 
1 Hectolitre . . . = , 

Weights. 

ENbl.ISH TROY. 

1 Grain (l-24th of a pennyweight) = 
1 Pennyweight (1 -20th of an ounce) = 
1 Ounce (1-1 2th of a pound troy) =: 
1 Pound troy imperial . = 

ENGLISH AVOIRDUPOIS. 

1 Drachm (l-lOth of an ounce) . = 

1 Ounce (l-16th of a pound) = 

1 Pound avoirdupois imperial = 

1 Hundred- weight (112 pounds) = 

1 Ton (20 hundred-weight) . = 



FRENCH. 



FRENCH. 

0-567932 litre. 

M35864 litre. 

4-54345794 litres. 

908G9I59 litres. 
36-347CG4 litres, 

1-09043 hectolitre. 

2-907813 hectolitres. 
1308516 hectolitres. 

ENGLISH. 

1-760773 pint 
0-2200967 gallon. 
2-2009667 gallons. 
22-009667 gallons. 



FRENCH. 

0*06477 gramme. 

1*55456 gramme. 

. 31-0913 grammes. 

0-3730959 kilogramme. 

FRENCH. 

1-7712 gramme. 
. 28-3384 grammes. 

0-4534148 kilogramm*. 
. 50-78246 kilogiammm. 
1015-649 kilogrammes. 

ENGLISH. 



1 Gramme 



Kilogramme 



15-438 grains troy. 
0-643 pennyweight. 
0032 16 ounce troy. 
2-68027 pounds troy. 
2*20548 pounds avoirdupoil 

Angular Measure, 

In France, the centesimal division of the circle is frequently em 
ployed, in which the whole circumference is divided into 400 degre« 
each degree into 100 minutes, each minute into 100 seconds, &c. 

CENTESIMAL. ENGLISH. 

1 Degree . . . , = . 54 minutes. 
1 Minute . . . = . 32*4 seconds. 

1 Second . . . . = . *324 second. 



ENGLISH. 



1 Degree 
1 Minute 
1 Second- 



CBNTESIMAL. 

1^ degree. 
1-85185 minute. 
8*08641 seconds. 



CHAP. IV.] MBOTUJEUTION OF SUPBRFICIE8. 12? 

Sect. II, Mensuration of Superficies* 

TRIANGLES. 

Let by c, and dy represent the three sides of a triangle (see 
fig- 95); 0j y, and ^, the angles opposite those sides respectively; 
and h the perpendicular height from the vertex to the hase b; 
then the 

area = | & A, 
or = I ^ c . sin ^ = ^ c (/ . sin jS = ^ </& . sin 7. 

Or, = a/ </^ — f — — H - ) • 2 where b is the greatest 

side and c the least 

If half the sum of the three sides or = », then the 



area = V» {s — A) . (a — c) . (« — d)y and the 
log of area =i{log« 4- log (« — i) -f log(« — c) -|- log (« — «?)}. 

For the method of obtaining the unknown sides or angles of 
triangles from those which are known, see Chap. V., Sect. IV. 

QUADRILATERALS, OR FOUR-SIDED FIGURES. 

Square^ Rectangle^ Rhombus and Rhomboid: — To obtain the area, 
naltiplj the perpendicular height (^, fig. 96) by the base {b) on 
which it falls. 

The area of a Trapezium is best found by dividing it into two 
triangles (as fig. 17), the areas of which may be found by the 
foregoing rules. 

The area of a Trapezoid (fig. 16) may either be found in the same 
way, or by multiplying half the sum of the two parallel sides (a -f b) 
by the perpendicular distance between them (Ji). 

POLYGONS. 

To obtain the area of any regular polygon, having less than twelve 
sides, multiply the square of one of the sides by the number found 
in colamn a, of the Table of Polygons, page 118. Or, generally, if 
/ =: the length of one of the equal sides, and n the number of them; 
then the 



area = /^ •- tan 

4 



^90 n— ISOX 



The area of an irregular polygon may be determined by dividing 
it into triangles, as in fig. 97. Or by forming one triangle equal in 
area to it, by Prob. XX., page 116. 



CIRCLES. 



Let d represent the diameter, e the circumference, a the area, 
mnd/» := 3*14159 (see Table V. in the Appendix) ; then the 



128 IfBNSURATION OF 8UPBRFICIB8. [PART I. 

_. , c 4a ^ /a 

Diameter =:(/ = - = — =2 a/~« 
p c ^ p 

4a 
Circumference = c =/? {/ = — = 2 ^p a. 

Area = « ='^,^ = /- = ^' = 'TSS*./'. 
4 4/? 4 

^ "" 5 "~ rf* "~ 4a' 

Circular Arcs. — If r represents the radius, and d the diameter of 
the circle; a the sine, and r, the versed sine of the arc; c the chord 
of half the arc, v^ the versed sine of half the arc; and m the mea- 
sure in degrees of the whole arc ; then 

the length of the arc = -0174533 rm; 



°'=2rf^^^^^ nearly; 



8c — 2« 
or = nearly. 

And the following relations between the several quantities, firom 
which any one of them may be obtained, will be found useful, viz.: — 

c, =r-N/i^^=7 .' (1.) 

c=n/ ^^H-^^ (2.) 

«= N/r'^-(r-t?J* (3.) 

^=f + t^x (*.) 

Circular Sectors. — Let d represent the diameter of the circle, 
/ the length of the arc of the sector^ and m its measure in degrees; 
then the 

area = | r /. 
or = -00218 J^w. 

Circular Segments, —Let d represent the diameter of the cirdc, 
© the versed sine, c, the chord of the whole arc, and c^ the chord of 
half the arc; then the 

area = | {-^{dv — ©^j ^ ^ ^dv} nearly; 
or = 1 1? (</«? — 1^ c^) nearly; 
<>r =^a<^(<^i +T<^2) nearly; 
or = -JtJ N/(i<?i^ + iv^) nearly. 

* Table VI. in the Appendix contains the diameter, circumference, area, and 
length of the side of an equal square to circles from 1 to 100*76 in diameter. 



CHIP. IV. j MENSUBATION OF 8UPBBFICIE8. 1 29 

Or the area of any segment, as fig. 98, may be obtained by finding 
(lie area of the sector A BCD, and subtracting from it tbe area of 
the triangle A C D. 

Tbe area of a circular zone^ as £ F G H, may be found by subtract- 
ing the area of the segment GH I from tbe area of tbe segment 
E6IHF. 

PABA30LA. 

The area of a parabola is equal to two-tbirds of the product of tbe 
l«e X the perpendicular height. 

For the relations between die abscissee and ordinates in tbe para- 
bola, and the method of deriving one from the other, see Chap. VI. 

page 172. 

Psrahdic Arcs, — Let x be the abscissa a 5, ^g, 99, measured from 
the vertex a, and y the corresponding right ordinate cb; then the 
length of the btc cadis 

= ^y/(f-ri^) nearly; 



= |{V(/ 



-h*^)-^ 



4 



y 



Parabolic Frustunij or Zone, — The word zone is here used to 
clcDotethe space ede^ (fig. 99), contained between the two parallel 
doable ordinates cd and e^. Let ^j = the length of the double 
^'fliMte ed^ and y^ = the length of e^y and d = the perpendicular 
datance between them bf; then the 



S ..3 



area of the zone = 4 ^ ^-^, ^^ • 

ELLIPSE. 

The area is equal to the product of the transverse diameter x the 
wnjngate diameter x 785398. 

'^periphery or circumference is equal to the sum of the trans- 
'eweand conjugate diameters x 1*57079. 

The foregoing is only an approximation to the periphery, although 
wfidently near the truth for ordinary purposes ; but where greater 
■ewracy is required the following series may be employed. Let t 
'^P'ttcnt the transverse axis, c the conjugate, p = 3* 14159, &c., 

«d <^= 1 — -; then 
r 

^j, d 8d' SK5d^ 3\5\7d* 

win be the periphery. 

EBipOe Segments, — To find tbe area of tbe elliptic segment abc 
(6g. 100), find tbe area of the corresponding circular segment, dbe^ 
Meribcd on the same axis (bf) to which tbe cutting line or base of 

K 



130 IfBNSUBATION OF 80LID8. [PAB 

the segment ae is perpendicular. Then, as this axis (hf) : 
other axis (ph) :: the circular segment (dbe) : the elliptic 
ment (a be). 

Elliptic Arcs. — Let t represent the semi-transverse, and e 
semi-conjugate diameters of the ellipse, and d the distance of 
ordinate from the centre, then the length of the arc conta 
hetwecn the ordinate and the parallel semi-axis will be 

-'^'''e?^'^-iO^'''' 112?^ ^ + *^ 

f^ — c^ 
or make ^ — = r ; then the 

length of the arc = €? ^ / —5 — . ^ nearly. 

HYPERBOLA. 

The area of an hyperbola or hyperbolic segment may be founc 
follows : — let t and c represent the semi-transverse and semi-coi 
gate diameters, 2y the double ordinate which cuts off the segmt 

and a its abcissa ; also g = : then the 

area = 2:»yfi -1 ?1 ^ &c} 

^^^ 3.5 3.5.7 3.5.7.9 ^ 

or =-£^(4 ^(2tx 4. f ar^) 4- ^Ztx] nearly. 

Hyperbolic Arcs. — The notation being the same, the follow 
approximation may be employed to obtain the length of an hy] 
bolic arc. 

120c-^ + (19/* + 21c')4« 
^'^ = 120c^/-K9f^ + 21c*)4^ +y' ^«^^y- 



Sect. III. Mensuration of Solids. 

PARALLELOPIPEDON, PRISM, OR CYLINDER. 

Surface. — Multiply the perimeter of one end by the lenstl: 
perpendicular height, to which add the area of the two en^ 
sum will be the surface. 

Solidity = the area of the base x the perpendicular height 

Pyramid or Cone. 

Surface. — Multiply half the perimeter by the slant height, to w 
add the area of the base, the sum is the surface. 

Solidity = the area of the base x one-third the perpendic 
height. 



CBihlV.] IfBNSUaATION OF SOLIDS. 131 

Pnutumofa Pyramid or Cone, Surface, — Multiply half the sum 
of the perimeters of the two ends hy the perpendicular height. 

SdidHy. — ^Add a diameter or side of the greater end to one of the 
leas, ind from the square of the sum subtract the product of the said 
two ditroeters or sides ; then multiply the remainder by one-third of 
the height, and this product by '785398 for circles, or by the proper 
Doitiplier for polygons ; the last product will be the capacity. 

That is, let D equal the greater diameter, d the less, h the perpen- 
^Iw beight, and p = "785398 for cones, or for any pyramid the 
proper multiplier from column (a) in the table at page 118, then 
Solidity = jjoA(D* + Drf + rf«). 

SPHBBB. 

S^irface = the diameter x the circumference, = the square of 
the diimeter x 3*14159, = the square of the circumference 
X •3183. 

Sdiditp = the cube of the diameter x '5236, = the cube of the 
cbmifercnce x 01688. 

8fkencd Segment. — Let d equal the diameter of the sphere, r 
the ndius of the base of the segment, and h its height ; then the 
otnednafaee = 3*14159 <f A. 

Solidiiy = 05236 k^ (3 </ — 2 A) ; 
or = 0-5236 A (3 r' -j- h"). 
The surface and solidity of a spherical zone may be obtained by 
^ing the difference between the two segments. 

CONOIDS. 

A conoid is the solid generated by the revolution of a conic section 
Jj^nt one of its axes, and is called a epheroidy paraboloid^ or hyper- 
^*fc*rf, according to the section from which it is produced. 
. Spheroids, — When the ellipse revolves about its transverse axis, it 
*• called an oblong or prolate spheroid; when about its conjugate axis, 
^'^ (Hate spheroid; and when about any -other of its diameters, a 
^^ivemd spheroi^ij in which latter case its figure is somewhat re- 
^^ttbling a heart. To obtain the solidity of a spheroid, multiply the 
?|^I^ttre of the revolving axis by the fixed axis, and the product 
'^ 0-5236. 

Pmdfoloid. — Let y be the radius of the circular base, and x the 
^titode of the solid ; then 

Surface = ^i^^ {(/ + 4:r»)? - y^}. 

Solidity = V5708y^x. 

y^HptrboUnd, — Let t equal the transverse axis, r the radius of the 
^*^ « the altitude, and p the parameter ; then the 



Solidity = J /^^ • . ^ 



K 2 



132 



MENSURATION OF SOLIDS. 



[PARr 



THE REGULAR OR PLATONIC SOLIDS. 

The regular or Platonic bodies are five in number, and have all 
been described at page 111. 

1. To find either the surface or the solid content of any of M 
regular bodies. — Multiply the proper tabular area or surface (talf:^ 
from column (a) in the following table) by the square of the lin ^ 
edge of the solid, for the superficies. And 

Multiply the tabular solidity in column {h) of the table by the cvsl 
of the linear edge for the solid content. 

Surfaces and Solidities of Regular Bodies^ the side being unity or 1 



No. of 
sides. 


Name. 


Surface. 


Solidity. 
(*) 


4 

6 

8 

12 

20 


Tetraedron 

Hexaedron 

Octaedron 

Dodecaedron 

Icosaedron 


1-7320508 
60000000 
3-4641016 
20-6457288 
8-6602540 


01178513 
1-0000000 
0-4714045 
7-6631189 
2-1816950 



2. The diameter of a sphere being given^ to find the side of 4^ 
of the Platonic bodies^ that may be either inscribed in the sphere '9 
circumscribed about the sphere^ or that is equal to the sphere, — ^^ 
tiply the given diameter of the sphere by the proper or correspond* 
number, in the following table, answering to the thing sought, ^^ 
the product will be the side of the Platonic body required. 



^ 



The diam. of a 
sphere being 1 ; 
the side of a 



Tetraedron 

Hexaedron 

Octaedron 

Dodecaedron 

Icosaedron 



That may be 
inscribed in the 
sphere, is 



That may be cir- 
cumscribed about 
the sphere, is 



0-8164966 
0-5773503 
0-7071068 
0-3568221 
0-5257309 



2-4494897 
1-0000000 
1-2247447 
0-4490279 
0-6615845 



That it equal 
to the sphere. 



1-64394A0 
0-8059958 
10356300 
0-4088190 
0-6214433 



3. The side of any of the five Platonic bodies being given^ to find 
the diameter of a sphere^ that may either be inscribed in thai body^ or 
circumscribed about it^ or that is equal to it, — As the respective 
number in the table above, under the title inscribed^ eircumscribedy 
or equcdy is to 1, so is the side of the given Platonic body to the 
diameter of its inscribed, circumscribed, or equal sphere. 

4. The side of any one of the fioe Platonic bodies being given^ to 
find the side of the other four bodies, that may be equal in solidi^ to 
that of the gioen body. — As the number under the title equal in the 



:aiP. IV.] MENSURATION OP SOLIDS. 133 

mMl colomn of the table above, against the given Platonic body, is to 
be number under tbe same title, against the body whose side is 
^agfat, 80 is tbe side of the given Platonic body to the side of the 
M)dj sought. 

Besides these tbere are thirteen demiregular bodies, called Solids 
^Archimedes, Tbey are described in the Supplement to Lidonne's 
f^M$ de tons les Diviseurs des Nombres^ &c., Paris, 1808 ; twelve 
»f them were described by Abraham Sharp, in his Treatise on 
E^olycdra. 

re nND THE CONTENTS OF SURFACES AND SOLIDS NOT REDUCIBLE TO 
▲BY KNOWN FIGURE, BY THE EQUIDISTANT ORDINATE METHOD. 

The general rule is included in this proposition, viz. : — If any 

ri^btline be divided into any even number of equal parts, 1, 2, 3, 4, 

&C., (fig. 101,) and at the points of division be erected perpendicular 

oxdiDates 1 A, 2 B, 3 C, &c., terminated by any curve A C G : then, 

iC a be put for the sum of the first and last ordinates, 1 A, 7 O, e for 

die sum of the even ordinates, 2 B, 4 D, 6 F, &c., viz., the second, 

fourth, sixth, &c., and o for the sum of all the rest, 3 C, 5 E, &c., 

"VIZ., the third, fifth, &c., or the odd ordinates, excepting the first and 

la«t: then, the common distance 12, 2 3, &c., of the ordinates 

l>cing multiplied into the sum arising from the addition of «, four 

^TDes e, and twice o, OTte third of the product will be the area 1 A G 7, 

▼wy nearly. 

_ / # + 4^ -h 2o 

That 18, — - — --^ . D = area, D being = A C = CE, &c. 

3 

The same theorem will equally serve for the contents of all solids, 
by oaog the sections perpendicular to the axis instead of the ordi- 
iistet. The proposition is quite accurate, for all parabolic and right 
^ areas, as well as for all solids generated by the revolutions of 
^ic sections or right lines about axes, and for pyramids and their 
^Wims. For other areas and solidities it is an excellent ap- 
proximation. 

The greater the number of ordinates, or of sections, that are taken, 
^ more accurately will the area or the capacity be determined. 
Bot b a great majority of cases^t^e equidistant ordinates, or sections, 
^ lead to a very accurate result. 



134 PLANE TRIOONOMBTRY. [PAKT I. 



CHAP. V. 

TBiaONOMETRY. 
Sect. I. Definitions and lVi</onotnetriciU Formulae. 

1. Plane Trigonometry is that branch of mathematics by which 
we learn how to determine or compute the unknown parts of a 
plane, or rectilinear triangle, from those which are known, when that 
18 possible. 

Every triangle consists of six parts, viz , three sides, and three 
angles opposite those sides. And any three of these being given 
(excepting only when the three angles are given) the others may 
always be determined from them. 

The determination of the mutual relation of the sincs^ tangents, 
secants^ &c., of the sums, differences, multiples, &c., of arcs or angles; 
or the investigation of the connected formulae, is also usually classed 
under plane trigonometry. 

2. Let ACB (fig. 102) be a rectilinear angle: if about C as a 
centre, with any radius C A, a circle be described, intersecting C A, 
CB, in A, B, the arc AB is called the measure of the angle ACB. 

3. The circumference of a circle is supposed to be divided or to 
be divisible into 360 equal parts, called degrees; each degree into GO 
equal parts, called minutes; each of these into 60 equal parts, called 
seconds; and so on to the minutest possible subdivisions. Of these, 
the first is indicated by a small circle, the second by a single accent, 
the third by a double accent, &c. Thus, 47° 18' 34" 45''', denotes 
47 degrees, 18 minutes, 34 seconds, and 45 thirds. The number of 
degrees, minutes, seconds, &c., contained in the arc AB of the circle 
described from the angular point C, and which is contained between 
the two legs AC, A B, is called the measure of the angle ACB, 
which is then said to be an angle of so many degrees, minutes, 
seconds, &c. Thus, since a quadrant, or quarter of a circle, contains 
90 degrees, and a quadrantal arc is the measure of a right angle, a 
right angle is said to be one of 90 degrees. 

4. The complement of an arc is its difference from a quadrant, 
as B E ; and the complement of an angle is its difference from a right 
angle, as £ C B. 

5. The supplement of an arc is its difference from a semicircle, 
as A' £ B ; and the supplement of an angle is its difference from two 
right angles, as A' C B. 



CHIP, v.] PJLANB TBIOONOIIBTKY. 135 

6. The tine of an arc is a perpendicular let fall from one extre- 
nitj opon a diameter passing through the other, as B D. 

7. the versed eine or versine of an arc is that part of the dia- 
eeter which is intercepted between the foot of the sine and the arc, 
uDA. 

8. The tangent of an arc is a right line which touches it in one ex- 
tRmity, and is limited by a right line drawn from the centre of the 
drde through the other extremity, as A T. 

9. The tecant of an arc is the radial line which thus limits the 
tugent, as C T. 

10. These are also, by way of accommodation, said to be the sine, 
tasgent, &c., of the angle measured by the aforesaid arc, to its deter- 
mintte radios. 

11. The cosine of an arc or angle, is the sine of the complement 
of that arc or angle, as G B : the cotangent of an arc or angle is the 
tiDgent of the complement of that arc or angle, as E M. And the 
OHxntd sine E O, and cosecant CM, are similarly the versed sine 
and secant of the complement. 

12. The suversine of an arc is the versed sine of its supplement, 
MAD. 

^flte.— The following contractions are employed to express the 
foregoing terms, viz. : — 

For the radius of the arc A B we write rad A B, 

sine ditto sin AB, 

tangent ditto tan AB, 

secant ditto «^c A B, 

versine ditto versin AB, 

cosine ditto om AB, 

cotangent ditto cot AB, 

cosecant ditto cosec AB, 

coversine ditto covers AB. 

13' The /oiiowing Corollaries may he drawn from the above 
Definitions: — 

(A.) When the arc is evanescent, the sine, tangent, and versed 
■Be^ are evanescent also, and the secant becomes equal to the radius, 
that being its minimum limit. As the arc increases from this state, 
the snes, tangents, secants, and versed sines increase ; thus they con- 
tnme till the arc becomes equal to a quadrant A £, and then the sine 
K m its maximum state, being equal to the radius, and is then called 
^ sine total; the versed sine is also then equal to the radius ; and 
^ tecaot and taneent becoming incapable of mutually limiting each 
<Hher, are regarded as infinite. 

Id employing these lines for the purposes of calculation, they are 
■D eonsidercKl as htmng positive values for any arc not exceeding 90®; 
hit in the second quadrant, the cosine falling on the opposite side of 
the diameter, and being measured in an opposite direction, is con- 
'i^ered negative^ but the sine remains positive; in the third quadrant 



136 



PLANE TRIGONOMETRY. 



[part I. 



the cosine is still negative^ and the sine, having now changed its direc- 
tion, is negative also ; in the fourth quadrant, the cosine having again 
returned to that side of the diameter on which it was in the first 
quadrant, again hecomes positive^ hut the sine remains neg€Uive. The 
signs of the others are determined by the ordinary rules of algebra, 
from the formulae at page 1 37. The following table exhibits both the 
value of trigonometrical lines at the commencement of each qua- 
drant, and also the signs with which they are affected in passini 
through the same. 





Value 
atO«. 


Sign 
in Ist 
Quad. 


Value 
atOO*. 


Sign 
in 2nd 
Quad. 

+ 


1 
Value' 
atl80*. 
_ 
O 


Sign 
in 3rd 
Quad. 


Value 

at 
270*. 


Sign 
in 4th 
Quad. 


Value 
at 

aeo*. 


Sin ... 


O 


-f 


R 





R 





o 


Tan... 


O 


-f 


00 


— 


o + 


00 


— 


o 


Sec ... 


R 


+ 


00 


- 


R — 


00 


+ 


R 


Versin 


O 


+ 


R 


■f 


2R + 


R 


+ 


o 


Cos ... 


R 


+ 


O 


- 


R 


— 


O 


+ 


R 


Cot ... 


00 


+ 


O 


— 


« 


-i- 


O 


— 


00 


Cosec 


00 


+ 


R 


-f 


00 - 1 


R 


— 


00 



R signifies equal to rad ; oo — infinite ; — evanescent. 

(B.) An arc and its supplement have the same sine, tangent, anc 
secant. 

(C.) Of any arc less than a quadrant, the arc is less than its cor- 
responding tangent; and of any arc whatever, the chord is less thai 
the arc, and the sine less than the chord ; but the smaller the arc, th< 
nearer they all approach to equality. 

(D.) The sine BD of an arc A B, is half the chord BF of th< 
double arc B A F. 

(£.) The versed sine of an arc, together ^4th its cosine, are equa 
to the radius. Thus, AD -fBG=AD + DC = AC. 

(F.) The radius, tangent, and secant, constitute a right-angle< 
triangle CAT; and the cosine, sine, and radius, constitute anothei 
right-angled triangle C D B, similar to the former. So, again, the co- 
tangent, radius, and cosecant, constitute a third right-angled triangle 
MEG, similar to both the preceding. Hence, when the sine anc 
radius are known, the cosine is determined by the property of the 
right-angled triangle. The same may be said of the determination ol 
the secant, from the tangent and radius, &c. &c. &c. 

(G.) Further, since the triangles CAT and MEC are similar 
A T : CA : : B D : D C : : C E : E M, or tan ; rad : : sin : cos : : rad 
: cot • 

Also C T : C A : : C B : C D : : C M : £ M, or sec : rad : : rad 
. : cosec : cot. 

And CM:CE::CB:DB or cosec : rad : : rad : sin. 



cos 



CHAP, v.] PLANS TBIGONOMBTBY. 187 

(H.) Also, by Geom. Sect. V. Prop. XXII., as AD : DB : : DB 
: HD, or sin* = H D . versin ; or since H D = pad -|- cos, tben 

sin^ 
tersin = 



rad + cos 

(I.) From Arts. (F.) (G.) and (H.) we deduce the following 
fonnolsB :— 

1. sin = >/Tad* — cos* = . 

cosec 



2. tan := >/sec* — rad* = 



3. sec = v^rad^* + tan* = 

cos 



rad . sin rad^ 
cos cot 

rad» 



4. versin = 



rad + cos * 



5. cos = >/rad'^ — sin* = 

sec 



6. cot := V cosec' — rad""* = 



rad« 
c 
rad . cos rad^ 



sin tan 



7. cosec =: Vrad* + cot* = -;— . 

sin 

cos* 

8. covers = • 



rad + sin* 



CK.) If unity be regarded as tbe radius of the circle, the above 
fonnoijg become : — 

1. sin = >/l — cos* = . 

cosec 

/ — 5 sin 1 

2. tan = V sec* — 1 := — = — - . 

cos cot 



3. sec = >/l + tan*= — . 

cos 

sin* 

4. versm = 



1 + cos 



5. cos = >/ 1 — sin* := — . 
sec 



cos 



cot = V cosec? — 1 = -r-^ : — 
sm tan 



138 PL^NB TBIOONOMBTBY. [PABT I. 

7» cosec = >/l + cot* = — r- . 

sm 

COS* 

8. covers = :-. 

1 + sm 

14. The following are some of tbe most useful formulsB relating 
to trigonometrical lines, in all of which radius is regarded as unity. 

(a.) Eapressions/or the sine and cosine : — Let a equal tbe length 
of the arc, then 

"'"* = ''-r:T:3 + 1.2.3.4.5 - 1.2.3.1.5 6.7 +^-^'^ 
oosa=l-^+P^^L-^- ^^ /^^^ +&c (2). 

(6.) Expressiofis for the sum and difference of two arcs: — Let 
a and h be the two given arcs, then 

sin (a 4- 5) = sin a cos 6 + cos a sin b (3) 

sin (a — 5) = sinacos b — cosasin b (4) 

cos (a 4- &) = cos a cos 5 — sin asin b (5) 

cos (a — ft) = cos a cos ft + sin a sin 6 (6) 

, ,. tan a + tan ft . 

tan(a -f ft) = , --^ — t (7) 

^ ' 1 ■— tan a tan ft ^ 

tana — tan ft 

tan(a-ft = — — — j- (8) 

"^ I -H tan a tan ft ^ ' 

cot a cot ft — i 

cot(a + ft) = TTT^ — 7— (9) 

^ ' cot ft + cot a ^ ^ 

„ cot a cot ft + 1 , ^ 

cot(a - ft) = -r -— (10). 

"^ ' cot ft — cot a ^ ' 

(c.) Expressions for the sine and cosine of mtdtiple arcs: — Let 
n be any integral number, then 

sin «a = cosa{(2sina)-» -^^(2sina)-^ + (!!Z?K^i:D(2sina)-^ 

- """!°.i.y"^ '-"A^<^c) ,„) 

cosnasz: J{(2sina)" — n(2sina)— * + ^ ? "l^V ^sina)-^ 

n(n — 4)(« — 5V . , . , 
1.2 3 ^^(28ina)-« + &c.} (12 



CHAP, v.] 



PLANl TBIOONOMBTBY. 



139 



(c^) Table o/mMkipk arcs : — Let « be the ain, t the tan, and e the 
cot of the arc a, then 



1. sin 



2"V 2 > 



2. 8ina=:«, 

3. 8m2a = 2« v^l — «*, 

4. m3a = Bs—4f\ 

5. 8in4a = (4* — 8r^ >/l — r*^ 

6. 8m5a = 16#' — 20*' + 5«, 



13.tan- = -'_, 

2 n.>/i^.^' 



7. cos 



2' 



'5. tan 2a = 

^^ Unda = 

^^- tan4a = 

^®- tan 5a = 



2/ 

1-3/** 
4^ — 4/» 



1— 6/» + /*' 
1 — 10/« + 5^' 



8. cos a ^ Vl — a", 

9. cos 2a = 1 —2a', 

10. cos 3a = (1 — 4«^) Vl— r', 

11. co84a = l — 8*«^-8«^ 

12. cos5a-(l-12**+16«*)Vr^. 

« 1 

19. cot- = , , 

20. cot a =e, 



21. cot2a=: 

22. cot 3a = 

23. cot 4a = 

24. cot 5a = 



c^-1 

c* — 3<? 
3c«- 1' 
c*^6c'+ 1 

4(r'-4c ' 
c^— 10c3 -i- 5c 
5c*— 10c« + 1 " 



5s. 



C ^') Table of Powers of the sine and cosine. 



un as una, 
Sain'aa 1 .-.oo«2a, 
liin^a"" Ssina — sin So, 
Sun* a« 3 — 4 COS 2a 4- COS 4 a, 
)6uii*as sin 6 a— 6 sin 3a4- lOsin a, 



6. cos aBCOsa, 

7. 2008*0-* 0062a + I9 

8. 4oos'a»co8 3a + 3oosa, 

9. 8G08*a»oos4a + 4oos2a + 3, 
10. 16co8*aB0066a-f5oos3a-i-10co8a. 



». ^/) Expressions for the arc in terms of the sine or tangent : — 
^^t « be the sin, and t the tan of the arc a, then 

(13) 



tKa = t — - + -—6ie 



(14). 



SscT. II. Trigonometrical Tables. 

From the forcing, and other properties, and theorems, mathe- 
ba¥e computed the lengths of the sines, tangents, secants, 



140 



PLANE TRIOONOMBTBY. 



[p^ 



.t 1 



and versed sines (assuming unity for the radius) corresponding to w 
from 1 second of a degree, through all the gradations of magni^^ad 
up to a quadrant, or 90^ ; and the results of the computation^^ t 
arranged for use in tahles called Trigonometrical Tablet. As, B=ioi 
ever, these quantities have to he carried to several places of decSL mi 
in order to ohtain sufficiently accurate results, their use in cal ^cali 
tions is attended with much lahour, and therefore it is usual to enr~^plc 
their logarithms instead ; hut in this case the assumed radius is t-^ce 
as 10,000000000 instead of unity, since with the latter most o^^ th 
quantities would he fractional^ and therefore have negative ch wuwt, 
teristics, the use of which would be inconvenient, and is supers^e 
by taking the radius as above. 

Table IV. in the Appendix is such a table of the logarithiKBS < 
the sines, cosines, tangents, cotangents, secants, and cosecants^ f< 
every minute from 1 minute to 90 degrees, calculated to a radius < 
10,000000000 as above. It will be observed that the headm^s * 
the columns run along the tops of the pages as far as the 45th de^i"^ 
after which they return along the bottoms of the pages in contra 
order, as below : — 



sin 


D. 


cosec 


tan 


D. 


cot 


sec 


D. 


co« 


cos 


D. 


sec 


cot 


D. 


tan 


cosec 1 D. 


&C^ 



The reason of this will be apparent, if we only consider that the cc^ 
cot, or cosec of an arc, is the sin, tan, or sec of the complement ^ 
that arc. The intermediate columns, headed D, contain the diffe^ 
ences of the consecutive logarithms in the contiguous columns o '^ 
either side ; it will be seen that the same difference is common to thti^ 
sin and cosec, the tan and cot, and the sec and cos; since fron^ 

rad* rad*^ 

Art. (I.) page 137, sin = — —^ or rad^ = sin . cosec; tan = — -^ 



or rad ^ = tan . cot ; sec = 



cosec 
rad 2 



cot 



cos 



or rad '^ = sec . cos ; and conse- 



quently, log sin + log cosec s= log tan 4" log cot = log sec + log 
cos = 2 log rad = 20* ; therefore as the sin, tan, or sec increases, 
so must the corresponding cosec, cot, or cos, diminish, and their dif- 
ferences must be equal. 

USB OP TABLE IV. 

To find the logarithmic sine^ tangent^ Spe, of a given are. 

If the arc contains only degrees and minutes, its sin, tan, &c., will 
be found simply by inspection, by looking along the top or bottom of 
the tables for the degrees, and then in the first or last vertical column 
for the minutes, according as the number of degrees is less or greater 
than 45 ; and on the same line, in the column having for its title 



.v.] PLANE TRIOONOMBTRT. 141 

ler At the top or bottom, according as the degrees were found) the 

z of the trigonometrical quantity required, its log will be found. 

f the arc contains seconds as well, the logarithm must be found as 

ife for the degrees and minutes ; then take the number in the 

otigooas column headed D on the same line, multiply it by the 

nnberof seconds, and divide by 100 (which is done by cutting off 

le two list figures) ; the quotient must then be added to or subtracted 

RND the log already taken out, according as the same would be in- 

aeued or decreased by an increase in the arc. 

Examples, 

1. Find the log sin of 37° 47'. 

As the arc is less than 45°, by looking along the top of the table for 
the degrees, and in the^r«/ column for the minutes, we find in the 
colomn baying at its top the word sin, the figures 9*787232, which is 
the log sin of the arc required. 

2. Find the log tan of 75° 34'. 

Here, as the arc is greater than 45°, looking at the battom of the 
tihles for the degrees, and in the last column for the minutes, we find 
in tbe column having tan at the bottom, 10*589431, the tan of 
rS'' 34'. 

3. Find the log sin of 31° 45' 5". 

Tie log sin of 31° 45' is . . . . 9-721 162 I 
The No. in col. D is 3402 x 6" -f- 100 = + 1 70 | 10 

.-. The log sin of 31° 45' 6" = . 9-721332 



4. Find the log cos of 25° 1' 47". 

Tbelogcos of 25° 1' is .... 995721 7 

The No. in col. D is 983 x 47" h- 100= ^ 462 01 



•. The log cos of 25° 1' 47" = . . 9956755 



Bb 



Tojind the are corresponding to any given log sin, tan, &c. : — 
^km Table IV. for the given log sin, &c., or the next less log 
^''^'^ and on the same line will be found the minutes, and at the 
^P or bottom of the page the degrees of the arc required ; if the log 
^ foond is less than the given log, subtract the former from the 
p^i add two cyphers to the right of the remainder, and divide it 
7 the nnmber found in the contiguous column headed D ; the quo- 
^Qt will he the nnmber of seconds to be added to the degrees and 
^tes in the arc already obtained. 

Examples. 

J* Fbd the arc wboae log tan is 10*577537. 
Here the arc is found by inspection to be 75° 11'. 



142 PLANB TBIQONOMBTBY. [PABT I. 

2. Find the arc corresponding to the log sin 9*395401. 

Given log == 9-396401 
Next less log = 9-396166 = log tan 14*> 23' 

23500 -f- 492 = 48 seconds ; 
.-. 9-396401 is the log tan of 14° 23' 48", 

In tbe sines and tangents of arcs less than about 2°, the differences 
between any two successive values are so great (as will be seen by 
an inspection of column D in tbe table), that the method above given 
for finding the intermediate values for seconds will not be sufficiently 
correct ; and the same remark applies to the cosines and tangents of 
arcs greater than about 88°. It will also be observed, that in the 
cosines and secants of arcs less than 2°, and in the sines and cosecants 
of arcs greater than 88°, the differences are too small to enable us to 
calculate accurately the value of any arc from them. 

The first of these difficulties may be removed by the rules given 
below for determining the values of the sines and tangents of small 
arcs, and the tangents of large arcs, and conversely the arcs from the 
sines and tangents. The second difficulty, however, could only be 
got over by extending the tables to more decimal places, but as this 
would also require all other Quantities employed in the same calcula- 
tions to be taken to an equal number of decimals, much additional 
trouble would be occasioned ; and it is therefore better for determin- 
ing the value of an arc when near 90°, to employ some other function 
than its sine, as, for instance, its cosine. In order to render this clearer 
to those who are not familiar with the use of logarithms, we subjoin 
an example of such a substitution of the cosine for the sine. 

For instance, let it be desired from the formula, 
Pj : Pg • : sin /9 : sin ^ 
to determine the value of the angle ^, when P j = 600-1 ; P^ = 669; 
and the angle = 63° 45'. 

First by multiplving the means and extremes, and dividing both 
sides by P, , we obtain 

. . Pnsinfi 
sm ^ = p (a). 

Then from Art. (L), page 137, 



sin ^ = >/ 1 — (cos iy 
Pj sin $ 



= N/l-.(cosd)«; 

•^1 



squaring both sides 

/ p, sin e y 

{^-^ — J = 1 - (cos ay, 



CKJIP.?.] PLANB TBIOONOMBTRT. 143 

trmssponng, and eztnctiiig the square root 



cos 



,=V-(^)- 



(«)• 



We haYe, therefore, two equations (a) and (^), from either of 

'vrlieh we can ohtain the yalae of the angle ^, hy substituting the 

iralQes of the known quantities ; but in doing so we shall find that 

the second equation will give the yalue of J much more exactly than 

tliefint. 

Thua^by tubstitnting the known quantities in equation (a), we have 

669 X sin 63° 45' . . 

-— — = sm i. 

6001 

Whence by logarithms. 

Log sin ea** 45' =r 9-952731 
Log 669 = 2-825426 

12-778157 
Log 600-1 = 2-778224 

Log sin i = 9-999933 

.-. i = 88<> 59' 25". 
Phiceedmg in a similar manner with equation (5), we have 

/669 X sin 63'' 45'V 

^-C — 6001 — ;=<^-^- 

Wbeoce by logarithms, taking the radius as unity, for the reasons 
wpliined at page 146, 

Log sin 63° 45' = 1952731 



Log 669 


= 2-825426 


Log 600-1 


2-778167 
= 2-778224 




T-999933 
2 


Loff -99969 


= r999866 



Then I ^ -99969 = -00031, the log of which = 4-491362, 
and 4-491362 -r 2 = 2-245681 = cos i; 
or restoring the radius of the tables, 

cos ^ = 8-245681 = sin of the complement of i ; 



144 PLINB TRIOONOMBTBY. [PABT I. 

therefore, bj the rule given below for finding a small arc from its sin, 
we have 

8-246681 

6-314425 

22 

3-560128 = 3631-86 seconds = r 0' 31" -86 ; 

^^=^ . ^ _ ggo 59/ 2g// .14^ 

whence we see that the former value of J obtained from equation 
(a) is upwards of three seconds too small. 

To find nccurately the log sin of an are less than 2°. 
Reduce the arc to seconds, and find the log of that number from 
Table III., to which add 4-685575 (the log sin of 1'), and subtract 
one-third of the decimal portion of the log sec of the arc taken from 
Table IV. ; the remainder is the log sin of the arc required. 

Examples. 
Find the log sines of 13' and of 1° 3'. 



Log (13' X eW) = 780" - 2-892096 
4-685575 



7-577670 
Log8ecofl3'»-000003-r3» -000001 



Log sin of 13' « 7-577669 



Log (ey X 60) « 3780" « 3-577492 
4-685575 



8-263067 
Logsecof r3'»-000073^3» -000024 

Log sin of 1° y ^ 8-263043 



To find accurately the log tan of an arc less than 2®. 
To the log of the number of seconds in the arc add 4-685575, and 
two-thirds of the decimal portion of its log sec; the sum is the log tan 
of the arc required. 

Examples, 
Find the log tans of 24' and of 1° l.V. 



Log (24' X 60} -: 1440" <= 3158363 

4-085575 

Log8ecof24'«*000011x}" '000007 



Log tan of 24' « 7843945 



Log (75^ X 60) - 4500" » 3-653213 

4-685575 

Log sec of ri5'»=-000l03x |» -000068 

Log Un of r 15^ « 8-338856 



To find accurately the log tan of an arc greater than about 88®. 

Add to the log of the number of seconds that the arc is less than 
90% two-thirds of the decimal portion of the log cosec, and subtract 
the sum from 15*314425; the remainder will be the log tan required. 

Examples. 
1. Find the log tan of 89° 5' 13". 
90'' 0' 0" 
89 5 13 



0^ 64' 47" = 3287 seconds 



CBAP.V.] PLANS TRIOONOMBTBY. 145 

Constant log .... = 15-314426 

Log of 3287 . . . ; = 3-6 16800 

Log cosec of 89« 5' 13'' = -000057 x | = -000038 

3-516838 



Log tan of 89'' 5' 1 3'' = 11797587 



a. Find the log tan of SS'' 61' 10". 
90° 0' 0" 
88 51 10 



r 8' 50" = 4130 seconds. 



CoMtMtlog =15-314425 

Ii0gof4130 =3-615950 

Xogcoscc of 88** 51' 10" = -000087 x | = -000058 

3-616008 



Log tan of 88° 61' 10" = 11-69 8417 

To find accurately an arc of not more than il°/rom its log sine, 
Totbegiyen log sin, add 5*314425, and one-third of the decimal 
poition of the secant of the nearest arc to that whose log sin is given, 
^« som, rejecting 10 from the characteristic, will be the logarithm 
^^ tke number of seconds in the arc. 

Example. 
Heqnired the arc whose log sin is 8314719, 

GiTen log sin . . . = 8-314719 

Constant log .... = 5-314425 
Ug sec of nearest arc = -000093 -f- 3 = -000031 

Arc required 1° 10' 58" = 4258" = 3-629175 

To find accurately an arc of not more than ^^from Us log tan, 
To the giTcn log tan add 5*314425, and from the sum subtract 
^''^[tJWs of the decimal portion of the log sec of the arc whose log 
^i« nearest to that given, and the remainder, rejecting 10 from the 
^''ctemtic, will be the log of the number of seconds in the arc. 

Example, 

Required the arc whose log tan is 8*231461. 

Given log tan . . . . = 8*231461 

Constant log . . . . = 6*314425 

13-545886 
Log sec of nearest arc = -000063 x | = -0000 42 

Required arc = 58' 34" = 3514" = 13-546844 

L 



146 GENEBIL PROPOSITIONS. [PABT I. 

To find (accurately an arc greater than S^"^ from its log tan. 

Add to the given log tan two-thirds of the decimal portion of the 
log cosec of the nearest arc to that whose log tan is given, and sub- 
tract the sum from 15*314425, the remainder is the log of the num- 
ber of seconds that the arc is less than 90^. 

Example, 

Required the arc whose log tan is 11*695900. 

Constant log ... . = 15-814425 

Given log tan .... =11-695900 
Log cosec of nearest arc 000088 X | = -000059 

11-695959 



Required arc = 1° 9' 14'' = 4154" = 3-618466 

ON THE ROOTS AND POWERS OF TRIGONOMETRICAL QUANTITIES. 

In extracting the root, or raising the power of any trigonometrical 
quantity by means of its logarithm, it will always be found most con- 
venient to reduce the assumed radius to unity, by subtracting 10 from 
the characteristic of the logarithm, observing where the characteristic 
thus becomes negative the rules given at pages 47 and 48 An ex- 
ample of this alteration of the charactenstic is given at page 143. 

TRIGONOMETRICAL QUANTITIES OP ANGLES GREATER THAN 90*^. 

Although Table IV. only purports to give the sines, tangents, &c., 
of angles less than, or equal to OO**, any of these functions of angles 
greater than 90° may readily be obtained, since any function of an 
angle greater than 90° is equal to the same function of the supple- 
ment of that angle ; due regard being had to changing the sign, if 
requisite, according to the table given at page 136. Thus the log sin 
of 141° 15' is 9-79052 1, the same as the sin of its supplement 
38° 45'; the log sec of 95° 43' is — J1001701, or the sec of 84° 17' 
with its sign changed ; the log cosec of the same is I0-00S165, or the 
cosec of 84° 1 7', the sign remaining the same ; and the log tan of 
173° 4' is — 9 084947, or the tan of 6° 56', with its sign altered. 



Sect. III. General Propositions, 

I. The chord of any arc is a mean proportional between the versed 
sine of that arc and the diameter of the circle. 

II. As radius is to the cosine of any arc, so is twice the sine of 
that arc to the sine of double the arc. 

III. The secant of any arc is equal to the sum of its tangent, and 
the tangent of half its complement. 

IV. The sum of the tangent and secant of any arc, is equal to the 
tangent of an arc exceeding that by half its complement. Or, the 



/ 



CfliP. v.] OBNBBAL PROPOSITIONS. 147 

9am of the tangent and secant of an arc is eqoal to the tangent of 
45'/>/i«half the arc, 

V. The chord of 60^ is eqaal to the radius of the circle; the 
^vereed sine and cosine of 60° are each equal to half the radius, and 
the secant of 60° is equal to double the radius. 

VI. The tangent of 45° is equal to the radius. 

VII. The square of the sine of half any arc or angle is equal to 
a rectangle under half the radius and the versed sine of the whole 
arc; and the square of its cosine is equal to a rectangle under half 
tiie radios and the versed sine of the supplement of the whole arc or 
aogle. 

nil. The rectangle under the radius and the sine of the sum or 
tile difference of two arcs is equal to the sum or the difference of the 
i^Bctangles under their alternate sines and cosines. 

IX. The rectangle under the radius and the cosine of the sum or 
^e difference of two arcs, is equal to the difference or the sum of 
the rectangles under their respective cosines and sines. 

X. As the difference or sum of the square of the radius and the 
'^^ctaogie under the tangents of two arcs, is to the square of the 
'^xiios; so is the sum or difference of their tangents, to the tangent of 
t^ sum or difference of the arcs. 

•XI. As the sum of the sines of two unequal arcs, is to their dif« 
^f^nce; so is the tangent of half the sum of those two arcs to the 
^Sigent of half their difference. 

^11. Of any three equidiffereut arcs, it will be as radius is to the 
^**«ne of their common difference, so is the sine of the mean arc, to 
"^If the sum of the sines of the extremes ; and, as radius is to the 
1^X1 c of the common difference, so is the cosine of the mean arc to 
"^If the difference of the sines of the two extremes. 

4^A.) If the sine of the mean of three equidiffercnt arcs (radius 
■^^ing unity) be multiplied into twice the cosine of the common dif- 
^^*^ce, and the sine of either extreme be deducted from the pro- 
^'ict, the remainder will be the sine of the other extreme. 

(B.) The sine of any arc above 60°, is equal to the sine of an- 
other arc as much below 60°, together with the sine of its excess 
•^Tc 60°. 

^. From this latter proposition, the sines below 60° being known, 
**^<»e of arcs above 60** are determinable by addition only. 

^ni. In any right-angled triangle, the hypothenuse is to one of 
|he legs, ag the radius is to the sine of the angle opposite to that 
^^; and one of the legs is to the other as the radius is to the tan- 
S^t of the angle opposite to the latter. 

. ^IV. In any plane triangle, as one of the sides is to another, so 
^ the sbe of the angle opposite to the former to the sine of the 
^^^ opposite to the latter. 

^^* In any plane triangle it will be, as the sum of the sides 
Jjj^t the vertical angle is to their difference, so is the tangent of 
'^ the sam of the angles at the base, to the tangent of half their 
«wrence. 

l2 



148 SOLUTION OP THK CASES OP PLANE TBIANOLBS. [PABT I. 

XVI. In any plane triangle it will be, as the cosine of the dif- 
ference of the angles at the base, is to the cosine of half their sum, 
so is the sum of the sides about the vertical angles to the third side. 
Also, as the sme of half the difference of the angles at the base, is 
to the sine of half their sum, so is the difference of the sides about 
the vertical angle to the third side, or base *. 

XVII. In any plane triangle it will be, as the base is to the sum 
of the two other sides, so is the difference of those sides to the dif- 
ference of the segments of the base made by a perpendicular let fall 
from the vertical angle. 

XVIII. In any plane triangle it will be, as twice the rectangle 
under any two sides, is to the difference of the sum of the squares of 
those two sides and the square of the base, so is the radius to the 
cosine of the angle contained by the two sides. 

Cor, When unity is assumed as radius, then if A C, A B, B C, are 

the sides of a triangle and C the angle opposite the side A B, this 

AC^ -h BC^ — AB- , . ., 

Prop, gives cos C = — - : and similar expressions 

<« C B . C A 

for the other angles. 

XIX. As the sum of the tangents of any two unequal angles is to 
their difference, so is the sine of the sum of those angles to the sine 
of their difference. 

XX. As the sine of the difference of any two unequal angles is 
to the difference of their sines, so is the sum of those sines to the 
sine of the sum of the angles. 

These and other propositions are the foundation of various for- 
mnlsB, for which the reader who wishes to pursue the inquiry may 
consult the best treatises on Trigonometry. 



Sect. IV. Solution of the C<we« of Plane Triangles, 
Although the three sides and three angles of a plane triangle, 
when combined three and three, constitute twenty varieties, yet they 
furnish only three distinct cases in which separate rules are required. 

CASE I. 

When a side and an angle are two of the given parts. 
The solution may be effected by Prop. XIV. of the preceding sec- 
tion, wherein it is affirmed that the sides of plane triangles are re- 
spectively proportional to the sines of their opposite angles. 

In practice, if a side be required, begin the proportion with a sine, 
and say, 

As the sine of the given angle. 

Is to its opposite side ; 

So is the sine of either of the other angles, 

To its opposite side. 

* Thene propositions were first given by Thacker in his Mathematical Miaeel- 
lanjfy published in 1743 ; their practical utility haM been recently shown by Pro- 
feMvr Wallac€t in the Edinburgh Philosophical TraruacHons. 






AV.Y.] SOLUTION OF TUB CASKS OF PLANE TRIANGLES. 149 

If an M^ he required, begin the proportion with a side, and 

As one of the given sides, 

Is to the sine of its opposite angle ; 

So is the other given side, 

To the sine of its opposite angle. 

The thud angle becomes known by taking the sum of the two 
former from 180°. 

NUe. — It is usually best to work the proportions in trigonometry 
bj means of the logarithms, taking the logarithm of the Jirst term 
from the sum of the logarithms of the 9econd and thirds to obtain the 
logarithm of the fourth term. Or, adding the arithmetical comple- 
ifiAi/of the logarithm of the first term to the logarithms of the other 
two, to obtain that of the fourth. 

CASE II. 

When two sides and the included angle are given. 

The solution may be effected by means of Props. XV. and XVI. 
of die preceding section. 

Thus: take Uie given angle from 180% the remainder will be the 
sum of the other two angles. 

TlienMy, 

As the sum of the given sides, 

Is to tlieir difference ; 

So is the tangent of half the sum of the remaining angles. 

To the tangent of half their difference. 
Then, secondly say, 

As the cosine of half the said difference. 

Is to the cosine of half the sum of the angles ; 

So is the sum of the ffiven sides. 

To the third, or required side. 

As the sine of half the diff*. of the angles, 
Is to the sine of half their sum ; 
So is the difference of the given sides, 
To the third side. 

^•tmjU.^hi the triansle ABC (Fig. 9.) are given A C = 450, 
^C 3 540, and the included angle C c= 80"* ; to find the third side, 
^ the two remaining angles. Then, 

Log(BC -AC as 90)= 1-954243 
Log tan (i A + B = 50<>) = 10076187 

12030430 
Log(BC-f'AC =: 990) = 2-995685 



Log tan (4 A - B =c 6^ 11") = 9034795 



150 SOLUTION OF THE CASES OF PLINE TRIANGLES. [PlBT I. 



Then, Log cos (| A -i- B = 50°) = 9-808068 
Log (BC -h A C = 990) = 2-995635 



12-803703 



Log cos (i A — B = 6° 1 V) = 9-997466 
Log(AB ... =640-08)= 2-806237 

Also, ^(A -i-B) + HA-B) = 56° ir=A; and J (A -f B) 
_ J (A - B) = 43° 49' = B. 

Here, much time will be saved in the work by talcing log cos J 
(A -h B) from the tables, at the same time with log tan ^ (A -h B) ; 
and log cos |(A — B) as soon as log tan ^ (A — B) is found. Ob- 
serve, also, that the log of B C -I- A C is the same in the second 
operation as in the first. Thus the tables need only be opened in 
Jive places for both operations. 

A nother solution to Case II, 
Supposing C to be the given angle, and C A, C B, the given sides ; 
then the third side may be found by this theorem, viz. ; — 

A B = ^/(A C^ -f B C'' — 2 A C . C B . cos C). 

Thus, taking A C = 450, B C = 540, C = 80% its cos -1736482 
AB = ^(450^^+ 540'^- 2 x 450 x 540 x -1736482) 
= ^{90=^ (5- + 6'- 2 X 5 X 6 X -1736482)} 
= 90 ^50-58118 = 90 X 7-112 = 64008, as before. 

CASE III. 

When the three sides of a plane triangle are given^ to find the angles, 

1st Method. — Assume the longest of the three sides as base, then 
say, conformably with Prop. XVI., last section, 
As the base. 

Is to the S!im of the two other sides ; 
So is the difference of those sides. 
To the difference of the segments of the base. 
Half the base added to the said difference gives the greater seg* 
ment, and made less by it gives the less ; and thus, by means of the 
perpendicular from the vertical angle, divides the original triangle 
into two, each of which falls under the first case. 

2nd Method. — Find any one of the angles by means of Prop. XVIII. 
of the preceding section ; and the remaining angles either by a repe- 
tition of the same rule, or by the relation of the sides to the sines of 
their opposite angles, viz. : — 

A C^ + B C2 — A B* A B'^ + B C — A C* 

cos C = ; cos B = ir7-~——^ 

2AC.BC 2AB.BC 

B A^ + A C^ — B C« 

and cos A = . 

2AB . AC 



r^HAP. v.] SOLUTION OP THB CASES OF PLANA TRIANGLES. 151 

RIGHT-ANGLED PLANE TRIANGLES. 

Right-angled triangles may (as well as others), be solved by means 
if the rule to the respecUye case under which any specified example 
alls; and it will then be found, since a right angle is always one of 
lie data, that the rule usually becomes simplified in its application. 

When two of the sides are given, the third may be found by 
neans of the property in Geom.^ Prop. XVI. Sect. III. 

Hypoth. = v^fbase^ -|- P^rp-^) 

Base = >/(hyp.* — perp.*) = >/(hyp. + perp.) . (hyp. — perp.) 

Pcrp. = ^(hyp.^ — base^) = >/(hyp. + base) . . (hyp. — hase). 

There is another method for right-angled triangles, known by the 
phnse making any Me radius ; which is this. 

** To find a side. Call any one of the sides radius, and write upon 
it the word radius ; observe whether the other sides become sines, 
tansents, or secants, and WTite those words upon them accordingly. 
Call the word written upon each side the name of each side: 
then say, 

As the name of the given side, 

Is to the given side ; 

So is the name of the required side, 

To the required side." 

^ To find an angle. Call either of the given sides radius, and 
vrite upon it the word radius; observe whether the other sides 
^^^come sines, tangents, or secants, and write those words on them 
accordingly. Call the word written upon each side the name of that 
"We. Then say, 

As the side made radius. 
Is to radius ; 

So is the other given side. 
To the name of that side, 
vhich determines the opposite angle." 

When the numbers which measure the sides of the triangle are 
<i^ nnder 12, or resolvable into factors which are each less than 
1^) the solution may be obtained, conformably with this rule, easier 
vithoQt logarithms than with them. For, 

Ut ABC (Fig. 103) be a right-angled triangle, in which A B, 
the hose, is assumed to be radius ; B C is the tangent of A, and A C 
te secant, to that radius ; or, dividing each of these by the base, we 
"ifl ha?e the tangent and secant of A, respectively, radius being 
^tj. Tracing in like manner the consequences of assuming BC 
(tt fig. 104), and AC (as fig. 105), each for radius, we shall readily 
•l>tMn these expressions. 

1. ^p^* = tan angle at base. (Fig. 103.) 

2. -^ = sec angle at base. (Fig. 103.) 
base 



152 HEIGHTS AND DISTANCES. [PART 

3. = tan angle at vertex. (Fig. 105.) 

4. -^-^ = sec angle at vertex. (Fig. 105.) 

5. -r—^' = sin angle at base. (Fig. 104.) 

base _, 

6. = sm angle at vertex. (Fig. 104.) 



Sect. V. On the application of Trigonometry to measuring HeighU 
and Distances. 

Trigonometry receives its principal practical application in the 
operations of surveying, and measuring heights and distances; as, 
however, the methods of its application (depending on the peculiar 
circumstances of each case) are exceedingly various, we cannot lay 
down any general rules, but must content ourselves with giving a 
selection of such examples as are most likely to occur ; and the prin- 
ciples developed in which, will be sufficient to guide any person in its 
further application to other cases. 

The instruments employed to measure angles are quadrants, sex- 
tants, theodolites^ &c., the use of either of which may be sooner 
learnt from an examination of the instruments themselves than from 
any description independently of them. For military men and for 
civil engineers, a good pocket sextant, and an accurate micrometer 
(such as Cavallo's) attached to a telescope, are highly useful. For 
measuring small distances, as bases, 50 feet and 100 feet chains, and 
a portable box of graduated tape, will be necessary. 

For the purposes of surveying, it is usual to employ a chain 66 feet 
in length, subdivided into 100 links, each 7*92 inches; the reason for 
using a chain of this length is, that ten of such square chains are 
equal to an acre, and therefore the acreage of the several divisions 
of an estate are found with much greater facility when measured in 
chains and links, than when the measurements are taken in feet. 

Eofample 1. 

In order to find the distance between two trees, A and B (Pig. 9), 
which could not be directly measured because of a pool which occu- 
pied much of the intermediate space, I measured the distance of each 
of them from a third object, C, viz., A C = 588, B C = 672, and 
then at the point C took the angle ACB between the two trees 
:= 55° 40'. Required their distance. 

This is an example to Case II. of plane triangles, in which two 
sides, and the included angle, are given. The work, therefore, may 
exercise the student: the answer is 593*8. 



CHiP.V.] HEIOHTS AND DISTANCES. 153 

Example 2. 

Wanting to know the distance between two inaccessible objects, 
C and D (Fig. 106), wbicb lay in a direct line from the bottom of a 
tower on whose top I stood, I took the angles of depreuion of the 
two objects, viz., of the most remote 25 J°, of the nearest 57°. 
^4t is the distance between them, the height of the tower A B 
being 120 feet? 
HAD = 25' 30', hence BAD = BAH — HAD = 64''30'. 
flAC=57'' O', hence BAG = BAH — H AC =33'' 0'. 
Hence the following calculation, by means of the natural tangents. 
^OT^ if A B be regarded as radius, B D and B C will be the tangents 
of tke respective angles BAD, BAC, and CD the difference of 
^ose tangents. It is, therefore, equal to the product of the difference 
»f tie natoral tangents of those angles into the height A B. 

Thus, nat. tan 64° 30' = 20965436 
nat tan 33° = 0*6494076 



difference 1-4471360 

multiplied by height, 120 

gives distance CD = 173*6563200 

*/ The natural sines, tangents, &c., are easily obtained from 
^^^We IV., by subtracting 10 from the characteristics, and then 
^iifiing the natural number answering to the logarithms with their 
^^^^•neteristics so altered. 

Example 3. 

Standing at a measurable distance AB (Fig. 103), on a hori- 
^t>l pltne, from the bottom of a tower, I took the angle of eleva- 
^ of the top (C) ; it is required from thence to determine the 
^l of the tower. 

In this case there would be given A B and the angle A, to find 
*C=:ABx tan A. 

B? logarithms, when the numbers are large, it will be log B C = 
^AB + logtanA. 



^«.-.If angle 


J A=ll° 19' 


then B C = ^ A B very nei 


» 


A = 16 42 




BC=AAB 


5> 


A = 21 48 




BC= 1 AB „ 


» 


A = 26 34 




B C = ^ A B „ 


99 


A = 30 58 




B C = 1 A B „ 


>9 


A = 35 




BC = 375AB 


99 


A = 38 40 




BC= 1 AB „ 


99 


A = 45 




B C = A B, ewacdy. 



To HYe the time of computation, therefore, the observer may set 
^ inttnunent to one of these angles, and advance or recede, till it 



154 HEIGHTS AND DI8TAKCSS. [PART I. 

accords with the angle of elevation of the object; its height above the 
horizontal level of the observer's eye will at once be known, by 
taking the appropriate fraction of the distance A B. 

Example 4. 

Wanting to know the height of a church steeple, to the bottom of 
which I could not measure on account of a high wall between me 
and the church, I fixed upon two stations at the distance of 93 feet 
from each other, on a horizontal line from the bottom of the steeple, 
and at each of them took the angle of elevation of the top of the 
steeple, that is, at the nearest station 5b° 54', at the other 33° 20^. 
Required the height of the steeple. 

Recurring to figure 106, we have given the distance C D, and the 
angles of elevation at C and D. The quickest operation is by means of 

the natural tansrents, and the theorem A B = -. 

® ' cot D — cot C 

Thus cot D = cot 33° 20' = 1*5204261 
C = cot 55 54 = -6770509 



Their difference = -8433752 

93 
"^"•"''^" = i433752 = "«-'''"^'- 

Eaoample 5. 

Wishing to know the height of an obelisk standing at the top of 
a regularly sloping hill, I first measured from its bottom a distance of 
36 feet, and there found the angle formed by the inclined plane and 
a line from the centre of the instrument to the top of the obelisk 
41°; but after measuring on downward in the same sloping direction 
54 feet farther, I found the angle formed in like manner to be only 
23° 45^ What was ttie height of the obelisk, and what the angle 
made by the sloping ground with the horizon ? 

The figure being constructed (see fig. 107), there are given in 
the triangle ACB, all the angles and the side AB, to find BC. 
It will be obtained by this proportion, as sin C (= 17° 15'= B — A) 
: AB (= 54) : : sin A(= 23° 45') : BC = 733392. Then, in 
the triangle D B C are known B C as above, B D = 36, C B D = 
41°; to find the other angles, and the side CD. Thus, first, as 
CB -f BD : CB — BD : : tan|(D -f C) = ^(139°) : tanJ(D — C) 
= 42° 24 J'. Hence 69° 30' + 42° 24|' =112° 54 J' = C D B, and 
69°30' — 42°24|' = 27°5J' = BCD. Then, sin BCD : BD :: 
sinCBD : CD = 51 '86, height of the obelisk. 

The angle of inclination DAE = HDA = CDB — 90° = 22° 54i'. 

Remark. — If the line BD cannot be measured, then the angle 
D A E of the sloping ground must be taken, as well as the angles 
C A B and C B D. In that case D A E + 90° will be equal to C D B: 



HEIGHTS AND DISTANCES. 155 

ifter C B is found from the triangle A C B, CD may be 
the triangle C B D, by means of the relation between the 
. the sines of their opposite angles. 

Example 6. 

; on a horizontal plane, and wanting to ascertain the height of 

standing on the top of an inaccessible hill, I took the angle 

•tion of the top of the hill 40% and of the top of the tower 

en measuring in a direct )ine 1 80 feet farther from the hill, I 

1 the same vertical plane the angle of elevation of the top of 

wer 33^ 45^. Required from hence the height of the tower. 

e figure being constructed (see fig. 108), there are given, AB 

«0, C A B = 33° 45', ACB = CBE — CAE=17°15', CBD 

r, BDC = 180° — (90° — DBE)=130°. And CD may be 

ad from the expression C D . rad^ = A B . sin A . sin C B D . cosec 

:B.8ecDB£. 



Or, 


Dsing logarithms. 
















log 


AB 


= 


180» 




= 


2-255273 




+ log sin 


A 


= 


33° 


45' 


= 


9-744739 




-f log sin 


CBD 


= 


11° 


0' 


= 


9-280599 




-f log cosec 


:ACB 


= 


17° 


15' 


= 


10-527914 




-f log sec 


DBE 


*"~ 


40° 


0' 


"~~ 


10-115746 




41-924271 




] 


log CD 


= 


log rad^ 
83-9983 


= 


40-000000 




1 924271 



Example 7. 

In order to determine the distance between two inaccessible ob- 
KcttAand B (Fig. 108), on a horizontal plane, we measured a con- 
TenieDt base, C D, of 536 yards, and at the extremities C and D took 
^following angles, via., D C B = 40^ 16', B C A = 57" 40', C D A 
=5 KV 22', A D B = 71° 7'. Required the distance A B. 

Rnt, in the triangle C D A are given all the angles, and the 
>^CD to find AD. So, again, in the triangle CDB, are given 
iQtlietDgles, and CD to find DB. Lastly, in the triangle DAB 
^ giren the two sides A D, B D, and the included angle A D B, to 
fad A B=i 939-52 yards. 

£mari.~ In like manner the distances taken two and two, be- 
^*^ soy number of remote objects posited around a convenient 
station line, may be ascertained. 

Example 8. 

Soppote that in carrying on an extensive survey, tbe distance be- 

^ecn two spires A and B (Pig. 109) has been found equal to 6594 

1^*^ and that C and D are two eminences conveniently situated 

for eitending ibe triangles, but not admitting of the determination 



156 HEIGHTS i\ND DISTANCB8. [PABT I. 

of tlicir distance by actual admeasurement: to ascertain it, therefore, 
we took at C and D the following angles, viz. : — 

CA C B = 85° 46' JA D C = 31° 48' 
(B C D = 23° 5G' |a D B = 68° 2' 

Required C D from these data. 

In order to solve this problem, construct a similar quadrilateral 
Acdhy assuming cd equal to 1, 10, or any other convenient number: 
compute A h from the given angles, according to the method of the 
preceding example. Then, since the quadrilaterals Acdh^ A C D B, 
are similar, it will be, as Ah : cd i: AB : Cl>; from which C D is 
found to be equal to 4694 yards. 

Example 9. 

Given the angles of elevation of any distant object, taken at three 
places in a horizontal right line, which does not pass through the 
point directly below the object; and the respective distances between 
the stations; to find the height of the object, and its distance from 
either station. 

Let A EC (Fig. 110) be the horizontal plane, FE the perpen- 
dicular height of the object above that plane, A, B, C, the three 
places of observation, FAE, FBE, FCE, the angles of elevation, 
and A B, B C, the given distances. Then, since the triangles A E F, 
BEF, CEF, are all right angled at E, the distances A£, BE, CE, 
will manifestly be as the cotangents of the angles of elevation at A, 
B, and C. 

Put AB = D, BC = </, EF = a?, and then express algebraically 
the theorem given in Geom. Sect. III. Prop. XXV., which in this 
case becomes, 

AE2.BC-hCE-.AB = BE2.AC-fAC.AB.BC. 

The resulting equation is 

dx^^QOiAf 4- Da?^(cotC)- = (D + rf)ar^(cotB)« + (D + d)l}d. 

From which is readily found 

(D-f fl?)P<^ 

rf(cot A)2 + D(cotC)^ - (D + <3?) (cot Bf ' 
Thus £ F becoming known, the distances A E, BE, C £, are 
found, by multiplying the cotangents of A, B, and C, respectively, 
by EF. 

Remark. — When D=</, or D-f-rf=2D = 2</, that is, when 
the point B is midway between A and C, the algebraic expres- 
sion becomes, 

a? = </-i. >/ 1 (cot A)* -h I (cot cy — (cot By, 

which is tolerably well suited for logarithmic computation. The rule 
inay, in that case, be thus expressed. 

Double the log cotangents of the angles of elevation of the ex- 
treme stations, find the natural numbers answen'ng thereto, and take 
half their sum ; from which subtract the natural number answering 



CHAP, v.] HSIOHTS AND DISTANCES. 157 

to twice the log cotangent of the middle angle of elevation : then 
haiftbelogof this remainder subtracted from the log of the mea- 
sured distance between the first and second, or the second and third 
stations, will be the log of the height of the object. 

The distance from either station \%ill be found as aboye. 

NtU.-The case explained in this example, is one that is highly 
useful, and of frequent occurrence. An analogous one is when the 
angles of elcTation of a remote object are taken from the three angles 
of a triangle on a horizontal plane, the sides of that triangle being 
known, or measurable : but the above admits of a simpler computa- 
tion, and may usually be employed. 

Example 10. 

From a convenient station P (Fig. Ill), where could be seen tliree 
objects, A, B, and C, whose distances from each other were known 
(ra. A B = 800, A C =-600, B C = 400 yards), I took the horizon- 
til ingles A P C = 33° 45', B P C = 22° 30'. It is hence required 
to detemine the respective distances of my station from each object. 

Here it will be necessary, as preparatory to the computation, to 
descnbe the manner of 

0(Mttnteti(m, — Draw the given triangle ABC from any convenient 
Kile. From the point A draw a line A D to make with A B an angle 
eqoal to 22° 30', and from B a line BD to make an angle DBA = 
33^45'. Let a circle be described to pass through their intersection 
D, ud through the points A and B. Through C and D draw a right 
line to meet the circle again in P: so shall Pbe the point required. 
For, drawing PA, PB, the angle A PD is evidently = AB D, since it 
^ds on the same arc A D : and for a like reason B P D = B A D. 
So that p is the point where the angles have the assigned value. 

The result of a careful construction of this kind, upon a good-sized 
««lc, will give the values of P A, P C, P B, true to within the 200th 
pwtofeach. 

Manner of Computation, — In the triangle ABC, where the sides 
*n known, find the angles. In the triangle A B D, where all the 
iiglesare known, and the sides A B, find one of the other sides A D. 
We B A D from B A C, the remainder, D A C, is the angle included 
Ween the two known sides, AD, AC; from which the angles ADC 
ttdACD may be found, by Case II., p. 149. The angle C A P = 
W - (A P C + A C D). * Also, BCP = BCA — ACD; and 
'BC = ABC H- PBA=:ABC-f- sup. ADC. Hence, the 
^^ required distances are found by these proportions. As sin 
APC: AC :: sin PAC : PC :: sin PC A : PA; and lastly, as sin 
*PC : BC : : sin B P C : B P. The results of the computation are, 
'A= 709-33, PC = 1042-66, PB = 934 yards. 

V The computation of problems of this kind, however, may be 
1 litde shortened by means of an analytical investigation. Those 
^ wish to pursue this department of trigonometry may consult the 
•"•*»es by Bmnycattky Gregory ^ and WwdhoxMe, 



158 HSIOHTS iIND DISTANCES. [PART I. 

Noie. — If C had been nearer to P tlian A B, the general principles 
of constmction and compotation would be the same; and the modi- 
fication in the process Terr obyioas. 

DBTSEMINATION OF HSIOHTS AND DISTANCES BY APPBOXIMATS ME- 
CHANICAL METHODS. 

1. For HeigkU. 

1 . By sAadatPS^ when the son shines. — Set op yertically a staff of 
known length, and measure the length of its shadow upon a horizon- 
tal or other plane ; measure also the length of the shadow of the ob- 
ject whose height is required. Then it w\]\ be, as the length of the 
shadow of the staff, is to the length of the staff itself; so is the 
length of the shadow of the object, to the object's height. 

2. By two rods or staves set up vertically: — Let two staves, one, 
say, of 6 feet, the other of 4 feet long, be placed upon horizontal 
circular or square feet, on which each may stand steadily. Let A B 
(Fig. 112) be the object, as a tower or steeple, whose altitude is 
required, and A C the horizontal plane passing through its base. Let 
C D and E F, the two rods, be placed with their bases in one and the 
same line C A, passing through A the foot of the object; and let tliem 
be moved nearer to, or farther from, each other, until the summit 
B of the object is seen, in the same line as D and F, the tops of the 
rods. Then bv the principle of similar triangles, it will be, as D H 
(= C E) : F H : : D G (= C A) : BG ; to which add A G = C D, for 
the whole height A B. 

3. By reflection, — Place a vessel of water upon the ground, and 
recede from it, until you see the top of the object reflected from the 
smooth surface of the liquid. Then, since by a principle in optics, 
the angles of incidence and reflection are equal, it will be as your 
distance measured horizontally from the point at which the reflection 
is made, is to the height of your eye above the reflecting surface ; so 
is the horizontal distance of the foot of the object from the vessel to 
its altitude above the said surface *. 

4. By means of a portable barometer and thermometer, — Observe 
the altitude B, of the mercurial column, in inches, tenths, and hun- 
dredths, at the bottom of the hill, or other object whose altitude is 
required ; observe, also, the altitude, 6, of the mercurial column at the 
top of the object ; observe the temperatures on Fahrenheit's thermo- 
meter, at the times of the two barometrical observations, and take the 
mean between them. 

* Leonard Digge§, in his curious work, the PatUometria^ published in 1571 « 
first proposed a method for the determination of altitudes by means of a geo- 
metrical square and plummet, which has been described by various later authors, 
as Ozanam, Donn, Hutton, &,c. But, as it does not seem preferable to the 
methods above given, I have not repeated it here. 



r.] HEIGHTS AND DISTANCES. 159 

B — i 

55000 X = height of the hill, in feet, for the tempera- 

of SS"" on Fahrenheit. Add ^ Jg of this result for every degree 
d) the mean temperature exceeds 55°; suhtract as much for every 
ree below 55°. 

rhis will he a good approximation when the height of the hill is 

« dian 2000 feet ; and it is easily remembered, because 55°^ the 

flomed temperature, agree with 55, the significant figures in the co- 

fident; while the significant figures in the denominator of the cor- 

.•sctiDg fraction are two /burs, 

5. Bifan extension of the principle of page 153.— Set the sextant, 
»r other instrument, to the angle 45°, and find the point C (Fig. 103) 
>zi the horizontal plane, where the object A B has that elevation : then 

set ibc bstrument to 26® 34', and recede from C, in direction BCD, 

kill the object has that elevation. 

7% Munce C D between the two stations will ^ = A B. 
So,sgain, if C = 40%D = 24°3lJ', CD will be = A B. 
or, „ if C = 35% D = 22° 23', CD „ = A B. 
or, „ if C = 30°, D = 20° 6', C D „ = A B. 
or, „ if C = 20°, D = 14° 5e\ CD „ = A B. 
or, generally, if cot D — cot C = rad, CD „ = A B. 

^- For deciation from lend. — Let E represent the elevation of the 
^^i>^t line to the earth above the true level, in feet and parts of a 
'^K D the distance in miles: then E = | D'^ 

Tbis gives 8 inches for a distance of one mile ; and is a near ap- 
P*^ximauoD when the distance does not exceed 2 or 3 miles. 

2. For Distances, 

1* By means of a rhombus set off upon a horizontal plane, — Sup- 

P^O(Fig. 113) the object and O B the required distance. With 

^ ibe or measuring tape, whose length is equal to the side of the in- 

^M rhombus, say 50 or 1 00 feet, lay down one side B A in the 

direction BO towards the object, and BC another side in any con- 

'^taJeni direction (for whether B be a right angle, or not, is of no 

^^'''^(tliience) ; and put up rods or arrows at A and C. Then fasten 

^ ends of two such lines at A and C, and extend them until the 

two other ends just meet together at D ; let them lie thus stretched 

*pOD the ground, and they will form the two other sides of the 

"•^^inbas AD, CD. Fix a mark or arrow at R, directly between C 

^ 0, upon the line A D ; and measure R D, R A upon the tape. 

'^ it will be as R D : D C : : C B : B O, the required distance. 

OdksTwise, To find the length of the inaccessible line Q R. 
At some convenient point B (Fig. 114) lay down the rhombus 
J^ADC, 10 that two of its sides, B A, B C, are' directed to the ex- 
^''"Wwi of the line Q R. Mark the intersections, O and P, of A R, 



160 HEIGHTS AND DISTANCES. [PART I. 

C Q, with the sides of the rhomhus (as in the former method) : then 
the triangle O D P will he similar to the triangle R B Q ; and the in> 

accessihle distance R Q will be found = — -.* 

OD X DP 

Thus,ifBA =BC,&c. =100fk.,OD = 9 ft. 5 ins.,D P = 11 ft. 
10ins.,OP = 13 ft. 7 ins., then QR = ^^t^-^^-~^ = 1219 feet. 

2. By means of a micrometer attached to a telescope. — Portable in- 
struments for the purpose of measuring extremely small angles, hare 
been invented by Martin, Cavallo, Dollond, Brewster, and others. In 
employing them for the determination of distances, all that is neces- 
sary in practice is to measure the angle subtended by an object of 
known dimensions, placed either vertically or horizontally, at the re- 
moter extremity of the line whose length we wish to ascertain. Thus, 
if there be a house, or other erection, built with bricks, of the usual 
size ; then four courses in height are equal to a fooi^ and four in 
length equal to a yard: and distances measured by means of these 
will be tolerably accurate, if care be taken while observing the angle 
subtended by the horizontal object, to stand directly in front of it. 
A man, a carriage-wheel, a window, a door, &c., at the remoter ex- 
tremity of the distance we \^ ish to ascertain, may serve for an ap- 
proximation. But in all cases where it is possible, let a foot, a yard, 
or a six-feet measure, be placed vertically, at one end of the line to 
be measured, while the observer with his micrometer stands at the 
other. Then, if h be the height of the object, 

either | A x cot \ angle subtended 
or A X cot angle 
will give the distance, according as the eye of the observer is hori- 
zontally opposite to the middle^ or to one extremity of the object 
whose angle is taken. 

When a table of natural tangents is not at hand, a very near ap- 
proximation for all angles less than half a degree^ and a tolerably 
near one up to angles of a degree^ will be furnished by the following 
rules. 

If the distant object whose angle is taken be 1 foot in length, 
then 

3437*73 -6- the angle in minutes \ will give the distance in 
or 206264- -^ the angle in seconds i feet. 

If the remote object be 3, 6, 9, &c., feet in length, multiply 
the former result by 3, 6, 9, &c., respectively. 

Ex. 1. What is the distance of a man 6 feet high, when he sub- 
tends an angle of 30 seconds ? 

• ForPD: DA:: AB: BR = ^^ ; andOD:OP::BR!RQ = 
A B' . O P 
O D . D P* 



CHAP, v.] HBI0HT8 AND DISTANCES. 161 

206264 X 6 -r 30 = 206264 -r 5 = 412528 feet =t= 137509 
yards, the distance required. 

Ex, 2. In order to ascertain the length of a street, I pat up a foot 
measure at one end of it, and standing at the other found that mea- 
sure to subtend an angle of 2 minutes : required the length of the 
street. 

3437-73 -r 2 = 1718-86 feet = 57295 yards. 

3. By means of the velocity of sound. — Let a gun be fired at the 
remoter extremity of the required distance, and obserre, by means of 
a chronometer that measures tenths of seconds, the interval that 
elapses between the flash and the report : then estimate the distance 
for one second by the following rule, and multiply that distance by 
the obserred int^val of time ; the product will give the whole dis- 
tance required. 

At the temperature of freezing, 32% the Telocity of sound is 1100 
feet per second ; for lower temperatures deduct, or for higher tem- 
peratures add, half a foot per second for every degree of difference 
from 32^ on Fahr. thermometer ; the result will show the velocity of 
sound, very nearly, at all such temperatures. 

Thus, at the temperature of 50% the velocity of sound is, 

1100 X i (50 — 32) = 1109 feet. 

At temperature 60% it is 1100 + i (60 — 32) = 1114 feet. 

For a more accurate method of determining the velocity of sound, 
I page 281. 



162 CONIC SECTIONS. [part I. 



CHAP. VI. 

CONIC SECTIONS. 
Sect. I. Dejiniti&ns* 

1 . Conic /Sections are the figares made by a plane cutting a 6one. 

2. According to the different positions of the cutting plane there 
arise five different figures or sections, viz., a triangUy a eirde^ an 
ellipse^ an hyperbola, and a parabola: of vt^hich the three last aie 
peculiarly called Conic Sections, 

3. If the cutting plaoe pass throngh the vertex of the cone, 
and any part of the base, the section will be a iHan^y as AB. 
(Fig. 115.) 

4. If the plane cut the cone parallel to the base, or make no angle 
with it, the section will be a circle , as A B. (Fig. 116.) 

5. The section is an ellipse when the cone is cut obliqnely through 
both sides, or when the plane is inclined to the base in a less angle 
than the side of the cone is, as A B C D. (Fig. 120.) 

6. The section is a parabola when the cone is cat by a plane 
parallel to the side, or when the cutting plane and the side of the 
cone make equal angles with the base, as P^ A P^ . (Fig. 138.) 

7. The section is an hyperbola when the cutting plane makes a 
greater angle with the base than the side of the cone makes, as 
PjAP,. (Fig. 130.) 

iVb/e.— In all the above definitions the cone is supposed to be a 
right cone. 

8. If all the sides of the cone be continued through the vertex, 
forming an opposite equal cone, and the plane be also continued to 
cut the opposite cone, this latter section is called the opposite hyper^ 
bola to the former, as S^ B S, . (Fig. 130.) 

9. And if there be two other cones, with their axes in the same 
plane and their sides touching the sides of the former cones, then 
will the same plane cut all the cones and form four hyperbolas, as 
P,AP«, R, CRo, S^BS^, Qj DQ^ (fig. 118), each opposite pair 
of which are similar ; these hyperbolas are said to be eonjtigate to 
each other, and the figure thus formed is called the figure of ike 
conjugaJte hyperbdas, as fig. 118. 

^(rfe.— In the following definitions, the letters refer to fig. 117 
for the ellipse; fig. 118 for the hyperbola; and fig. 119 for the 
parabola. 



CHAP. VI.] CONIO 8I0TI0N8. 103 

10. The veriice9 of any section are the points where the cutting 
pkne meets the opposite sides of the cone, as A and B. 

Cor, Hence the ellipse and the opposite hyperbolas hare each two 
vertices; but the parabola only one; unless we consider the other as 
at an infinite distance. 

1 1. The mqfor amt^ or iransverte diameter of a conic section, is 
the line or distance A B between the Yertioes. 

Cor. Hence the axis of a parabola is infinite in length, A B being 
ody a pari of it. 

1 2. The eenhre O is the middle of the axis. 

Cor, Hence the centre of a parabola is infinitely distant from the 
vertex. And of an ellipse, the axis and centre lie within the curve : 
bot of an hyperbola, without. 

Id. The mtHor^ or conjugate aariSj is the line C D, drawn through 
the centre perpendicular to the transverse axis, and bounded each 
Way by the curre. 

CJcr, Hence the parabola has not any conjnsate axis, unless we 
aappoee it at an infinite distance, and infinite in length. 

14. A Umgemi to a curve at any point, is a line as A^ T^ , which 
tooehes the carve in that point; but being produced either way docs 
not cut it. 

15. A diameter is any right line, as A^Bj, drawn through the 
csentre, and terminated on each side by the curve ; and the extremi- 
ties of the diameter, or its intersections with the curve, as A ^ and 
H J , are its vertices. 

Cor, Hence all the diameters of a parabola are parallel to the 
Axia, and infinite in length. Hence, also, every diameter of the 
ellipse and hjrperbola has two vertices; but of the parabola, only one; 
aaleaa we consider the other as at an infinite distance. 

1^. The coi^ugaU to any diameter is the line drawn throngh the 
centre, and pandlel to the tansent of the curve at the vertex of the 
diameitf. So, 0, D, , panllel to the tangent at A ,, is the conjugate 
toAjBji. 

17. An mrdimate to aay diameter is a line parallel to its conjugate, 
or to tlie tangent at its vertex, and terminated by the diameter and 
carve. So^ K I and K, Aj are erdinates to the axis A B ; and K, I 
•ad fi| I| ordiaates to the diameter A, B^ . 

Cor. ilesoe the ordinatea of the axes are perpendicular to it; but 
of other diameters, the ordinates are oblique to them. 

18. An oiirisfa is a pvt of any diameter, contained between its 
vertex and an ordinate to it; as AK or BK, and A, E, or B, S, . 

Cor. Hence, in the ellipse and hyperbola, every ordinate has two 
abec ioom ; \ml in the parabola only one; the other vertex of the 
diaeieter beiag infinitely distant. 

19. The ^wrtex from which the abcissa are measured is called the 
origin of those abcissse, and any abcissa and its ordinate are called 
co^frdinaieSi as AK and KI. 

20. The parameter of any diameter is that double ordinate which 
is a third proportional to the transverse and conjugate axes in the 

M 2 



164 CONIC 8BCTI0N8. [PART I. 

ellipse or hyperbola, and to any abscissa and its ordinate in the para- 
bola. The parameter of the transyerse axis is called the principal para- 
meter^ or the lattis rectum; thus, if A^ B. : C, D, :: Cj D, : I^ I^, 
then Ig I4 is i\\e parameter of A ^ Bj ; and if AB : CD :: CD : I'j I3, 
then I J I3 is the lotus rectum, 

21. The point where the parameter cuts the transyerse axis is 
called \he focuB^ as E, F; and the distance of the focus from the 
nearest yertex of the same axis is called the focal digtance^ as AE, BF. 

Cor, Hence, the elHpse and hyperbola haye each two foci, but the 
parabola only one. The foci (or burning points) were so called 
because all rays are united or reflected into one of them, which pro- 
ceed from the other focus, and are reflected from the curye. 

22. The point £ is also frequently called the poie; and a line 
drawn from any point in the curye to the pole, is called the radius 
vector y as E I. And the angle B £ I, contained between the radius 
yector and the transyerse axis, is called the traced angle, 

23. The directrix is a right line drawn perpendicular to the trans- 
yerse axis of a conic section, through an assignable point in the 
prolongation of that axis; such that lines drawn from any points in 
the curye parallel to the axis to meet the directrix, shall be to lines 
drawn from the same points to the focus, in a constant ratio for the 
same curye. 

Thus, if A E : A T :: £ I^ : I«y :: E I3 : I3 « :: E I : I X, then 
X Y is the directrix. 

In the ellipse A £ is less than A T. 

In the parabola A E is equal to A T. 

In the hyperbola A E is greater than A T. 

24. The subtangent to any point in a curye, is that portion of the 
transyerse axis which is contained between the tangent and ordinate 
to the same point, as K^ T, . 

25. A normal at any point is a line drawn from that point per- 
pendicular to the tangent, to meet the transyerse axis, as A ^ L. 

20. The Radius q/* Curvature of a conic section or other curye, is 
the radius of that circle which is precisely of the same cunratnre as 
the curye itself, at any assigned point, or the radius of the circle 
which fits the curye and coincides with it, at a small distance on 
each side of the point of contact. The circle itself is called the oseu- 
latorg circle, or the eguicurve circle ; and if the curye be of inces- 
santly yarying curyature, each point has a distinct eqaicarye circle, 
the radius of which is perpendicular to the tangent at the point of 
contact. 

27> An asymptote is a right line towards which a certain cunre 
line approaches continually nearer and nearer, yet so as never to 
meet, except both be produced indefinitely. The hjrperbola has two • 
asymptotes, as UV, WZ. 



CHir. VI.] CONIC SBCTIONS: ELLIPSE, 165 

Sbct. II. Properties of the Ellipse. 
1. Let the frostum of a right cone AGBH (fig. 120), he cut by 
apkne fonning a tangent to each end, then will the section (ACBD) 
thus prodaced be an ellipse. Let dszQBy the diameter of the lesser 
end; D = AH the diameter of the greater; A = AG, or BH, the 
dant height; t=iAB^ the transverse diameter of the ellipse ; c = C D, 
the eonjugile diameter; andysEF, the distance between the two 
foci; then, 

^=A2 + Drf (1.) 

c» = D^ (2.) 

/ =A (3.) 

1 Lety=IK (fig. 117), any ordinate; a;=:AK, its abscissa 
nettored from A ; ;y =: O K, its abscissa measured from the centre, 
O;0 = IE, the radiup vector from the focus £; i9= the traced 
»a^ IBF; and ^= >^i^'— i<^; then, 

y = f ^(/4r-««) (4.) 

viiich b the equation to the ellipse when the absciss® are measured 
ftm the vertex ; 

y=f Va<'-**) (5.) 

vbich is the equation when the abscissas are measured from the 
•tttre; and 

^^ it — g.coBS ^ '^ 

thich is called the polar equation^ and finds its principal use in the 
miei^mtions of astronomy. 

3. For the relation between the abscissse and ordinates in any 
•%ie, we have. 

As the square of the transverse azis^ 

: the square of the conjugate ; 

: : the rectangle of the abscissas, 

: the square of their ordinate. 

Or, /«:tf^::«(/-«):y» (7.) 

"Rw sune proportion obtains between the ordinate and abscissae to 
•^ditmetcr; or putting «, for Aj^j (fig. 117); w^ for B^Kj, 
^^y,forIKi; ^^^ 

t,*:e,^::w,x^:y,* (8.) 

^ tarn of the squares of any pair of conjugate diameters is 
^ to the sum of the squares of the two axes; or putting t^ for 
^i^ (fig. 117), and e^ for C^ D^ ; then 

t^ + e'^t^^^e,^ (9.) 

^U the parallelograms that can be circumscribed about an ellipse 
•reequil to the rectangle of the two axes; or = ct (10.) 



106 CONIO 8BCTI0N8: BLL1P8B. [pABT I. 

The sum of the two radius vectors, drawn from the two foci, to 
any point in the curve, are equal to the transverse axis; or putting 
e = £ I (fig. 117), the radius vector from £, and V = F I, the same 
from F ; we have 

V -hr = / (11.) 

Cor. The distance of either focus from either extremiljr of the 
conjugate diameter is ^oal to half the tmnsverae axis ; or, 

CE =CF = | (12.) 

The square of the distance between the two foci equals the differ- 
ence of the squares of the two axes ; or, 

r^t'^e' (1«.) 

4. tf TM (fig. 121) be the tangent to an ellipse at any point I, 
and let T be the point in which the tangent meets the transverse 
axis produced; also from the two foci draw FM, EN, perpendicular 
to the tangent, and produce E N and F I to meet each other in P ; 
then, 

OK : AO :: AO : OT; or, OT — -- (14.) 

Cor. If there be any number of ellipses described on the same 
transverse axis (fig. 122), and any ordinate be drawn, cutting all the 
ellipses in the points I^, I^, I^, I^, &c., the tangents to the several 
ellipses at those points, wilt all meet the transverse axis produced in 
one common point, T 0^') 

The angles made by tiie two radius vectors and the tangent arc 
equal; or, 

2LTIE=:/.LIF (16.) 

The points M and N (fig. 121) fall in the circumference of a circle 
Ivhose diameter is the transverse axis, AB (17.) 

Also, EN. FMarCO'tfc ^1 J (18.) 



And, EN* 



= (I)'t (•••) 



If m be in the middle of K I , , then, A m prodctced wOi meet the 
two tangents TQ, BQ, in their point of intersection, Q ... (20.) 

If the ordinate 1 1 ^ passes through the focus, tben the point T of 

intersection of the tangent and the transverse axis prodac^ will be 

a point in the directrix (21*) 

c' 
In any ellipse the parameter = — (^^0 

5. Let R ss the radius of curvature at any point in an ellipse, and 
V, «, the distances of the same point from the two foci; then 

R = *^ (23-> 



CBAf. YI.] COMIC MCTIOVS: BLLIP81. 167 

It is A mtunmum at the extremity of the conjugate axis, when 

*^-f. ("•> 

And a minimum at the extremity of the transverse axis, when 

»* = ^ (25.) 

The area of any ellipse is a mean proportional hetween the areas 
of the circles deambed on its two axes (^^0 



P90BLSM8 jaiLATINO TO THX ELLIPSE. 

^«OB. I. To find the two foci E^ Fy of an ellipse^ of which the 
transverse cune A B, and conjugaU C D, are given. 

Prom C or D AS a centre, with a radius equal to the semi- trans- 
^'cvce axis AC or BO, describe an arc cutting the transverse axis in 
^ J, wLich are the two foci required. (Fig. 123.) 

FiOB. II. To von^mct an ellipse whose two axes are given, 

Knd the distance £F, from Prob. I., and let a fine thread, £IF£, 
?* length = £F + A B, be put round two pins fixed at the points 
^ ^; theo, if a pencil be put within the cord, and carried round, 
^^>^ being taken diat the cord is alwavs tight, the pencil will dc- 
•*^*"ibe an ellipse, A I C B D A. (Fig. 1 24*) 

^^^^41. III. The transverse and conjugate amis of an ellipse being 
given^ to describe the same with a trammel, 

let the distance between the pencil A and the first pin B, be 
^^^ to half the conjugate axis, and the distance between A and 
^^^ leeond pin C, be equal to half the transverse ; then, the pins 
****iBg pot into the grooves of the trammel, and the pencil A being 
**^^>^ will describe the ellipse. (Fig. 125.) 

OQierwise, — ^Let there be provided three mlers, of which the two 

^^) FH, are of the same length as the transverse axis AB, and the 

^^vdHP, equal in length to £F, the focal distance. Then, con- 

^^^cting these rulers so as to move freely about £ F H and I^ their 

^•iimtiun I will always be in the curve of the ellipse : so that, 

^f tkere be alits mnning along the two rulers, and the apparatus 

^*ned fiteiy aboat the foci^ a pencil put through the slits at their 

^^ sf intenectaon will deacribe the carve. (Fig. 126.) 

Pbob. ly. To find the two axes of any given ellipse, 

^W tny two pamUel lines across the ellipse, as M L, F K : bisect 
"•"» in the points I and D, through which draw the right line 



168 CONIC sections: hypbrbola. [part i. 

N I E P, and bisect it in O. From O as a centre, with any adequate 
radius, describe an arc of a circle cutting the ellipse in the points 
G, H. Join O, H, and parallel to the line G, H, draw through O 
the minor axis CD; perpendicular to which through O draw AB, 
which will be the major axis. (Fig. 127.) 

pROB. V. From any given point out of an ellipse to draw a 
tangent to it. 

Let T be the given point, through it and the centre C draw the 
diameter A B ; and parallel to it any line H I terminated by the curve. 
Bisect HI in O ; and C O produced will be the conjugate to A B. 
Draw any line TS = TB, and make TR = TC. Draw RA, and 
parallel to it, SP cutting AB in P. Through P, draw PM parallel 
to CD, and join TM, which will be the tangent required. (Fig. 128.) 

Prob. VI. To find the length of an elliptic arclj C I^ . 

Produce the height CE to O, the centre of the ellipse. Join 
I J and O, and from O as a centre with radius C O eqttai the semi- 
conjugate, describe an arc cutting I , O in a. Bisect 1 1 a in 5, and 
from O as a centre with radius O b describe the arc bcy catting C O 
produced in c; then the circular arc ^c is equal in length to half the 
elliptic arc I, C Ig . (Fig, 129.) 



Sect. III. Properties of the Hyperbola. 

1. Let two opposite right cones (Fig. 130) be cut by a plane 
making a less angle with their axes than the sides of the cones make, 
then the sections thus produced will be two opposite hyperbolas. Let 
d = AHy the lesser diameter of the cone at the vertex of one hyper- 
bola ; D = B G, the greater diameter of the cone at the vertex of the 
opposite hyperbola ; /t = A G, or B H, the slant height ; / = A B, the 
transverse axis; c = CD, the conjugate axis; and /*= £F, the dis^ 
tance between the two foci ; then we have 

/«=:A2-Drf . . . (1.) 

c' = Drf . . . . (2.) 
f = h . . . . (3.) 

which will be seen to be identical with the corresponding equations 
for the ellipse, with the exception of the first, which, however, only 
differs in the sign of the last term. This at once indicates a generu 
analogy between the properties of the two curves ; and if we employ 
the same letters to represent the corresponding lines in figures llTi 
121, 122, and figures 118, 131, 132, the formula already given for 
the ellipse will apply to the hyperbola, only making the following 
alterations. 



CHAP. VI.] 4x>Nic sections: hypkbbola. 169 

(4.) Becomes y = T/V/ ^'* "'' *'^' 
(5) « 9=-fAji''-\n- 



(6.) „ f> = ^ 



I < + ^ • COS /3 

(7.) and (8.) remain unaltered. 

(9.) Becomes <,* — c^' = f* — c*. 

(10.) All the parallelograms that can be inscribed between four 
conjugate bjrperbolaSy are equal to the rectangle of the two axes ; or, 
= et 

(11.) Becomes V— © = ^ 

(12.) „ OEorOF = ACorCB. 

(13.) „ r^f-^e. 

(14.) to (25.), both inclusiTe, remain unaltered. 

2. Besides these, however, there are several curious properties 
which relate to the tuymptotes of the hjrperbola. Let 8), '3, 829 
(fig. 133) be an hyperbola; OV and OZ its asymptotes; 9^ K«^, 
a double ordinate to the transverse axis; O^r^, Oz^^ Oz^^ Oz^^ Oj»^, 
abscissa taken on the asymptote OZ; and;!;, 'i>^8 '^^ ^3 'si ^4 '49 
z^ «5, ordinates to the same, drawn parallel to the other asymptote 
OV: 

Then the parallelograms 0©^ *, ^r,, Ov^ 9^ z^y O©,, «, z^y &c. 
ire all equal ; or 

0*1 -^1 *i = ^^« • *2 *s =0^3 .^3 *j, =04r4 .«4«4 (26.) 

/« 

Also, m#i . <i *a = ««5 . «i 85 = ^*s' = J • • (27.) 

And the triangle O T I, (Fig. 131) = the triangle O B Q . (28.) 

Also, if the abscissae Oz^^Oz^^Oz^^ &c. of any h^rperbola, be 
taken on one of the asymptotes in an increasing eeometncal progres- 
sion, the ordinates z^9^y z^ ^^ , z^ 9^^ &c. parallel to the other asymp- 
tote are in decreasing geometrical progression, having the same 
imtio (29.) 

And, when the distances Oz^y Oz^^ &c. are in geometrical pro- 
gression, the asymptotic spaces z^ 9. So z^^ Zy «. 9^ jT,, &c. will be 
m arithmetical progression, and will, therefore, be analogous to the 
logarithms of the former (30.) 

The radix of the system of logarithms will depend upon the 
value of the angle made by the two asymptotes. In Napier 9 
logarithms V O Z is a right angle ; while in the common logarithms 
it is 24** 44' 254" ♦. 

* See Sect. 6, page 93. 



170 COXIC 8BCTI0K8: HYFIRBOLA. [PABT I. 

3. In the cmse of foar conjngate bypcrboIa8, the transyerse and con- 
jugate axes of one pair become the conjugate and transTerse of the 
other pair, and th^efore they are said to be matually conjugate to 
each other. 



PBOBLRMS RELATING TO HYPERBOLAS. 

Prob. I. To detcrihe an Hyperbola of tckich the tranwerM axis and 
twofod (ire given. 

Let one end of a long ruler, £0 (fig. 134) be fictttened at one of 
the foci, £, so as to turn freely about that point as a centre. Then 
take a thread F I , O, of such a length, that when one end is fixed at 
the other focus F, and the other end to the ruler at O, the doubled 
part will just reach to B, one end of the transverse axis. Then if the 
ruler E O be turned about the fixed point E, at the same time keep- 
ing the thread O I , F always tight, and its paK I , O close to the side 
of tlie ruler, by means of the pencil I j ; the curve line B S . de- 
scribed by the motion of the pencil I, is one part of an hyperbola. 
And if the ruler be turned, and move on the other side of the fixed 
point F, the other part B S^ of the same hyperbola may be described 
after the same manner. But if the end of the ruler be fixed in F, 
and that of the thread in £, the opposite hjrperbola P ^ A P, jao^y be 
described. 

Othencise; also by continued motion. Let E and F (Fig. 135) 
be the two foci, and A and B the two vertices of the hyperbola. 
Take three rulers, C D, D I, I F, and fix them so that E D s: G F ax 
A B, and DC =: EF; the rulers D I and 6 1 being of an indefinite 
length beyond I, and having slits in them for a pin to move in ; and 
the rulers having holes in them at £ and F, to fiasten them to the foci 
£ and F by means of pins, and at the points D and C they are to be 
joined by the ruler D C. Then, if a pencil be put in the slits, at the 
common intersection of the rulers D I and F I, and moved along, 
causing the two rulers F I, ID, to turn about the foci E and F, that 
pencil will describe the portion A P j of an hyperbola. 

Prob. II. To describe an hyperbola of which the two axes aregiveuy by 
finding points in the curve. 

Let A B be tlie transverse axis, and C D the conjugate. First from 
O as a centre, with radius equal A C or C B, describe arcs cutting 
the tuansverse axis produced in £ and F, which will be the two foci. 
Then assume any number of points, a^t a,, a^y &c. beyond F, and 
from £ as a centre, with radii equal to Ba,, Bao, Ba,, &c., describe 
arcs ^,9 ^^9 ^89 ^^* ' ^^^ ^^^^ F as a centre, with radii equal to 
Aa,, Aa.„ Aa^, &c., describe arcs cutting the former arcs in the 
points c^^c^^ C3, &c., which will all be points in the hyperbola re- 
quired. (Fig. 136.) 



CHAP. VI.] CONIC BJraTlONB: PABABOLA. 171 

Prob. III. To deteriht an kyperhokL, of which the transvene axis A B, 
and any abscissa A O, and its doubU ordinate H I, are given. 

Through A draw a h parallel to H I ; from H and I draw H a and 
I h parallel to A O ; then divide aH and &I, each into any number of 
equal parta, 1, 2, 3, &c., and through the points of division draw lines 
radiating to A as a centre ; also divide K H and K I into the same 
number of equal parts, and through the points of division draw lines 
nuliatiog to B, then will the several points «,, Cg, C3, &c, in which 
these lines intersect the former lines, be so many points in the 
hyperbola required. (Fig. 137.) 



Sbct. IV. Properties of the Parabola. 

1. If the right cone KHI (fig. 138) be cut by a plane parallel 
to the side KH, then will the section P^ A P,, thus produced, be a 
parabola. Let the cone also be cut by a plane, M A, perpendicular to 
the plane ^ the parabola, and by another plane, G A, jiarailel to the 
base of the cone, and both passing through the point A ; bisect the 
distance M O (between their two vertices) in L, and through L, take 
a plane L T, parallel to the base of the cone, and let S be the focus 
of the parabola. Let & = B P^, half the base of the parabola ; d = 
OA, the dinneter of the cone at the vertex of the paral)ola ; /= A E, 
ike focal diBtanoe ; and A s K A, the alant height of the cone above 
dieaame; iImo 

The Mm TX, in which the plane L T meets the plane of the para- 
bola, is its directrix ... ... (1.) 

d^ 
TA=/=lj (2.) 



b 



2 



f-Vd w 

2 (M O) = 4/ = the parameter . . (4.) 

2. Let /> = I^ I3 (fig. 119), the parameter of a parabola; 
« vs A &, aay aheeisaa ; jf bb I, X, the corresponding ordinate ; v = 
£ I, the radiiB vector; and /9 ss the tnced angle II B, £ being the 
focus; then 

f^px. . . . (5.) 

f> = , ^^ ^ . . (6.) 

1 q: cos g ^ ' 

the ayiitiopg to the paiabok : im the latter of which, or the polar 
eauation, the sign + obtains when K is between A and S, and — 
when K is below £. 

(4^ -f p\\ 
Rad. of curvature at I = „ ^^^ . (7.) 

2 >/;t> ^ 



172 CONIC sections: parabola. [part i. 

At the vertex, A, x vanishes, and we have rad. 

of curv. at vertex = i /> • • . (8.) 

3. In the same figure, where XY is the directrix, the following 
properties ohtain ; viz. 

AE = AT, EIj = Igy, EI3 =130:, EI =IX, &c.(9.) 

AsAK : AK, :: KI* : K.P 2; or ^^ = ^^ ^ v (10.) 

where A K and A R, are any ahscissss, and K I, K3 P ^ their corre- 
sponding ordinates. Also 

EI = AK + AE, EPj = AK3 4- AE . (11.) 

AE = ni=Lil (12.) 

Ij I3 heing the parameter = p, 

As/?:K,P^ + Kgl :: K,Pj - K,I :AK3 - AK, ^ . . 
or, asjt? : P^a :: aP^ : KgK3 i 

4. Again, let A , T ^ he the tangent to a parahola at any point A|, 
and let B ^ A J Y he drawn through A . parallel to the axis A B ; let 
A J L he perpendicular to A^ T. ; then is K, T, the suhtangent, A . L 
the normal, K^ L the suhnormaJ; and the following properties obtain; 
viz. 

angle EA, T^ = angle ET, A^ = angle T,Aj Y . . (15.) 

angle LAjBj = angle LA^E (1^0 

EA,= ETi (17.) 

AKj = ATj (18.) 

subtangent K, Tj = 2 AKg (19.) 

V 
subnonnal K, L = 2AE = ^a constant quantity . • (20.) 

5. In figure 139 also, where CQ is a tangent to the parabola at the 
point C, and IK, OM, QL, &c., parallel U) the axis AD. 

Then IE : EK :: CK : KL . . (21.) 

and a similar property obtains, whether CL be perpendicular or ob- 
lique to TD. 

The external parts of the parallels IE, TA, ON, QL, &c., are 
always proportional to the squares of the intercepted parts of the tan- 
gent; that is, 

the external parts IE, TA, ON, QL, \ 

are proportional to CI', CT', CO% CQ^ [ . . (22.) 

or to the squares CK^ CD^, CMV CL', ' 



CHAP. VI.] CONIC sections: pababola. 173 

And as this property is common to every position of the tangent, 
if the lines IE, T A, ON, &c., he appended to the points I, T, O, &c., 
of the tangent, and movahle ahout them, so as always to hang verti- 
cally, and of such lengths that their extremities E, A, N, &c., he in 
the curve of a parahola in any one position of the tangent ; then 
making the tangent revolve ahout the point C, the extremities E, A, N, 
&c., will always form the curve of some parahola, in every position 
of the tangent. 

The same properties, too, that have heen shown of the axis, and 
its ahscisses and ordinates, &c., are true of those of any other dia- 
meter. 

PROBLEMS RELATING TO THE PARABOLA. 

Pbob. I. To anutruet a Parahola^ of which the base and height are 

given. 

Construct an isosceles triangle ABD (fig. 140) whose hase AB 
shall he the same as that of the proposed parahola, and its altitude 
CD twice the altitude G V of the parahola. Divide each side AD, 
DB, into 10, 12, 16, or 20, equal parts [16 is a good numher, hecause 
it can he obtained by continual bisections], and suppose them num- 
bered 1, 2, 3, &c. from A to D, and 1, 2, 3, &c. from D to B. Then 
draw right lines I, 1; 2, 2; 3, 3; 4, 4; &c., and their mutual inter- 
section will beautifully approximate to the curve of the parahola 
AVB. 

Pbob. II. To describe a Parabola by finding points in the curve^ an ab- 
scissa A B, and its double ordinate C D, being given. 

Through A (fig. 141) draw a b parallel to CD ; and through C and 
D draw aC, 6D parallel to AB; then divide aC and bD into any 
convenient number of parts, 1,2, 3, &c., and through these divisions 
draw lines radiating to A ; also divide B C and B D into the same 
number of parts, and through the divisions draw lines perpendicular 
to CD, then the points c,, c^* c^, &c., in which these lines intersect 
the former, will be in the parahola. 

Let the ruler, or directrix BC (fig. 142) be laid upon a plane with 
the square ODO, in such a manner that one of its sides DO lies along 
the edge of that ruler; and if the thread F M O, equal in length to 
D O, (Uie other side of the square,) have one end fixed in the extre- 
mity of the ruler at O, and the other end in the focus F : then slide 
the side of the square D O along the ruler B C, and at the same time 
keep the thread continually tight by means of the pencil M, with its 
part MO close to the side of the square DO; so shall the curve 
A M X, which the pencil describes by this motion, be one part of a 
parabola. 

And if the square be turned over, and moved on the other side of 
the fixed point F, the other part of the Rame parabola A M Z will be 
described. 



174 CONIC 8BCTI0N8 : PARABOLA. [PART I. 

Pbob. III. Any right line being given in a parabola^ to find the corre- 
sponding diameter; aleoy the axis, parameter^ and focus. 

Draw HI (fig. 143) parallel to the given line DE. Bisect D £, and 
H I, in # and O, through which draw A O O for the diameter. Draw 
H R perpendicular to A O and hisect it in B; and draw V B parallel 
to A O for the axis. Make V B : H B : : H B : parameter of the 
axis. Then ^ the parameter set from V to F gives the focus. 

Prob. IV. To draw a tangent to a Parabola, 

If the point of contact C (fig. 144) he given, draw the ordinate, 
CB, and produce the axis until AT = A B : then join T C, which will 
he the tangent. 

Or if the point be given in the axis produced: take A B ^ AT, 
and draw the ordinate B C, which will give C the point of contact ; 
to which draw the line T C as before. 

If D be any other point, neither in the cunre nor in the axis pro- 
duced, through which the tangent is to pass: draw DE O perpendi- 
cular to the axis, and take D H a mean proportional between DE and 
DO, and draw H C parallel to the axis; so shall G be the point of 
contact, through which and the given point D the taD|*mit D C T is to 
be drawn. 

When the tangent is to make a given angle with the ordinate at tbe 
point of contact : take the absciss A I equal to half the parameter, or 
to double the focal distance, and draw the ordinate I £ : also draw 
A H to make with A I the angle A H I equal to the given angle; then 
draw H C parallel to the axis, and it will cut the curve in C the point 
of contact, where a line drawn to make the given angle with C B will 
be the tangent required. 



CHAF. VII.] CURVES. 175 



CHAP. VII. 
PROPBBTIER OP CURVES. 

Sect. I. Definitions. # 

1. A curve line^ as already defined, (Def. 5, page 98,) ia a line 
which continually changes its direction. 

2. Apiane carve, is one the several points of which are all situate in 
the Mme plane ; and a curve of double curvature is one whose several 
points are not aJl in the same plane, and which, therefore, curves in 
two directions. 

3. Curve lines have heen further divided into AlgebraiccU or Geo- 
metrieat^ and TrtxMtemiental or MeckanicaL 

4. An tdgehrauud curve, is one in which the relation which the 
abseista bears to the ordinate, taken for any possible point in that 
corye^ may always be expressed by a constant algebraical formula. 

5. A transcendental cuTvCy is one in which no such constant rela- 
tion between its abscissse and ordinates exists, and which can, there- 
fore, only be described by mechanical means. 

6. The algebraical formula which expresses this relation is called 
the equation of the curve; and curves are classed into orders, accord- 
ing to the number of dimensions contained in such equations. Thus 
any curve line which might be represented by the equation 

a: a + ^x -H cy + da^ + exy +/y*, 

would be called a line of the second order, because the abscissa {ai) 
and ordinate (y) are involved only to the second power; and a curve 
which might be expressed by 

= • + 6« + cy -h <^^ -h ««y +ff -h gas" + hw-y + hf» + If 

would be called a line of the third order, because the same quantities 
are involved to the third power. The letters a, &, c, &c., are merely 
the coefficients depending upon the particular curve expressed by the 
equation. 

7. An am/mpMe to a curve is a straight line, to which the curve 
line continually approaches, without ever meeting it; as the asymp* 
iotes U V and W Z, to the hvperbola, fig. 118. 

8. If a curve cuts itself by passing twice through the same point, 
as A, fig. 145, this point is called &punctum duplea: if three times, as 
K, it is called a punctum tripUx^ &c. 



170 curves: conchoid. [part i. 

9. The oval contained between A and C, fig, 145, is called a 
nodus: when the distance between A and C becomes indefinitely 
small, so that the nodus vanishes, the point A is called a punclum 
cofijugatum, 

1 0. A point in which two branches of a curve terminate, is called 
a cuspia, 

1 1 . The involute of any curve, is another curve traced by the end 
of a string in being unwound from the first curve, which is called the 
evolute. Thus, let A B C (fig. 1 46) be a curve having a thread laying 
in contact with every part of the same ; then, if the thread be un- 
wound from A, the curve line A D £« which its end will describe, 
will be the involute to the curve ABC, which latter will be the evo- 
lute of the <?tirve A D £. 

The principal lines connected with curves, such as tangenty ordinate^ 
absciBSoe^ &c., having been already defined in the first section of 
Chap. VI., it is unnecessary to recapitulate them here. 

12. The equations of the conic sections have been already given, 
for the ellipse at page 165, the hyperbola at page 169, and the para- 
bola at page 171 : the equation for the circle is 

y = s/dx — aj*, 

d being the diameter ; the whole of these four curves are therefore 
lines of the second order, the quantities a; and y, only being raised to 
that power. We shall now pass on to describe the principal pro- 
perties of those curves which are most frequently employed for useful 
purposes. 



Sect. II. The Conchoid. 

Conchoid^ or Conchiles^ (from the Latin concha^ a shell,) is the 
name given to a curve by its inventor, Nicomedes^ about 200 years 
before the Christian era. 

The conchoid is thus constructed: AP and BD (fig. 147) being 
two lines intersecting at right angles: from P draw a number of 
other lines PFDE, &c., on which make always DE = DF = AB 
or BC; so shall the curve line drawn through all the points E, £, E, 
be the first conchoid, or that of Nicomedes; and the curve drawn 
through all the other points, F, F, F, is called the second conchoid; 
though, in reality, they are both but parts of the same curve, having 
the same pole P, and four infinite legs, to which the line DBD 
(called the directrix) is a common asymptote. 

The inventor, Nicomedes, contrived an instrument for describing 
his conchoid by a mechanical motion, in the following manner: in the 
ruler DD (fig. 148) is a channel or groove cut, so that a smooth nail 
firmly fixed in the movable ruler CA, in the point D, may slide 
freely within it: into the ruler AP is fixed another nail at P, for the 
movable ruler AP to slide upon. If, therefore, the ruler A P be so 
moved as that the nail D passes along the groove D D, the style, or 
point in A, will describe the first conchoid. 



CHAP. VII.] 0URVB8 : CI880ID. 1 77 

CoDchoids of all possible Turieties may also be constructed with 
great facility by Mr. Jopling's apparatus for curves, now well known. 

1. Let AB = BC == DE = DF = a, PB = ft, BO = EH = x, 

and OE = BH = y : then the equation to the first conchoid will be 

«* (ft -I- «)* + xy = a- (ft + «)*, 
or, a?* + 2ftd^ + ft«a?' + x'f =^0^1^ + 2a'fta? + a^«*; 

and, changing only the sign of x^ as being negative in the other 
curve, the equation to the second conchoid will be 
a?' (ft - xY -h a?V = a^ (* - x)\ 
or, «* — Zhx" + h^x^ + x^'f = a" ft- — 2a-ftx + aV. 

2. Of the conchoids expressed by these two cauations (or rather 
the same equation with different signs), there are three species ; first, 
when a is lets than ft, the conchoid will be of the form shown in 
fig. 148, the inferior branch having a punctum conjugatum at A ; 
secondly, when a is equal to ft, the conchoid will assume the form 
shown in fig. 1 49, the punctum conjugatum giving place to a cuspU at 
B ; and thirdly, when a is greater than ft, the cuspis becomes a noduSj 
the conchoid taking the form shown in fig. 150. 

Newton approved of the use of the conchoid for trisecting angles, 
or finding two mean proportionals, or for constructing other solid 
problems. But the principal modem use of this curve, and of the 
apparatus by which it is constructed, is to sketch the contour of 
the section that shall represent the diminution of columns in ar- 
chitecture. 



Sbct. III. The Ciuoid or Cyewid. 

The eisMoid is a curve invented by an ancient Greek geometer and 
engineer named Diodes^ for the purpose of finding two continued 
mean proportionals between two given lines. This curve admits of 
an easy mechanical construction ; and is described very beautifully 
by means of Mr. Jopling's apparatus. 

At the extremity B of the diameter A B (fig. 151), of a given circle 
AOBo, erect the indefinite perpendicular eBE, and from the other 
extremity A draw any number of right lines, AC, AD, AE, &c., 
catting the circle in the points R, O, M, &c. ; then, if C L be taken 
= AR, DO = AO, EN = AM, &c., the curve passing through the 
points A, L, O, N, &c. will be the eissoid. 

1. Here the circle AOBo is called the generating circle; and AB 
is the axis of the curves ALON, &c., A/ow, &c., which form a 
cuspii at A, and, passing through the middle points O, o, of the two 
semicircles, tend continually towards the directrix eBE, which is 
their common asjrmptote. 

2. Letting fall perpendiculars LP, RQ, from any corresponding 
points L, R ; then is A P = BQ, and A L = CR. 



178 curves: cycloid. [pabt i. 

3. Also, AP : PB :: PL^ : AP*. So that, if the diameter AB of 
the circle = a, the abscissa A P = «, and the ordinate P L = y ; then 

« : a — jc :: y- : jr, or ar* = (a — *)y% 

which is the equation to the curve. 

4. The arc AM of the circle = arc BR, and arc Ant = Br. 

5. The whole infinitely long cissoidal space, contained between 
the asjrraptote ^ B E and the curves NO LA, &c., and A Lou, &c., is 
equal to three times the area of the generating circle AOBo. 



Sect. IV. The Cycioidy and Epicycloid, 

The cycloid^ or trochoid^ is an elegant mechanical carre first 
noticed by Descartes^ and an account of it was published by Jffr- 
senne, in 1615. It is, in fact, the curve described by a nail in the 
rim of a carriage-wheel while it makes one revolution on a flat 
horizontal plane. 

1. Thu8,if a circle EPF (fig. 152),keeping always in the same plane, 
be made to roll along the right line A B, until a fixed point P, in its 
circumference, which at first touched the line at A, touches it again 
at B after a complete revolution ; the curve APVPB described by 
the motion of the point P is called a cychid. 

2. The circle EPF is called the yeneraiing circle ; and the ri^t 
line A B, on which it revolves, is called the base of the cycloid. 

Also, the right line, or diameter CV, of the circle, which bisects 
the base A B at right angles, is the axis of the cycloid ; and the point 
V, where it meets the curve, is the vertex of the cycloid. 

3 If P rfig. 153) be a point in the fixed diameter AF produced, 
and the circle A E F be made to roll along the line A B as before, so 
that the point A, which first touched it at one extremity, shall touch 
it again at B, the curve PVP, described by the point P, is called a 
curtate cycloid. 

4. And, if the point P (fig. 154) be anywhere in the unproduced 
diameter A P, and the circle A EF be made to roll along AB from A 
to B; the curve PVP, is, in that case, called the inflected or prdaie 
cycloid. 

The following are the chief properties of the common cycloid. 

5. The circular arc VE (fig. 155) is equal to the line EG between 
the circle and cycloid, parallel to A B. 

6. The semi-circumference VEC is equal to the semi-base CB. 

7. The arc VG equals twice the corresponding choinl VE. 

8. The semi-cycloidal arc VGB equals twice the diameter VC. 

9. The tangent TG is parallel to the chord VE. 

10. The radius of curvature at V equals 2 C V. 

11. The area of the cycloid AVBGA is triple that of the circle 
CEV; and consequently that circle and the spaces VECBG, 
V E' C A Q\ are equal to one another. 



CHAP. VII.] curves: QUADRATRIX. 179 

12. A body falls through any arc LK (fig. 156) of a cycloid re- 
Teraed, in the same time whether that arc be great or small ; that is, 
from any point L, to the lowest point K, which is the vertex re- 
versed: and that time is to the time of falling perpendicularly through 
the axis M K, as the semi-circumference of a circle is to its diameter, 
eras 3-141593 : 2. And hence it follows that if a pendulum be 
made to vibrate in the arc L K N of a cycloid, all the vibrations will 
be performed in the same time. 

1 3. The eoolvte of a cycloid is another equal cycloid, so that if 
two equal semi-cycloids OP, OQ (fig. 156), be joined at O, so that 
O M is equal to M K the diameter of the generating circle, and the 
•tring of a pendulum hung up at O, having its length equal to O K or 
the curve OP; then, by plying the string round the curve OP, to 
whieh it is equal, if the ball be let go, it will describe, and vibrate 
in the other cydoid PKQ; where OP equals QK and OQ equals 
PK. 

14. The cycloid is the curve of swiftest descent : or a heavy body 
will fall from one given point to another, by the way of the arc of 
% cycloid passing through those two points, in a less time than by 
ma^ Olher route. Hence, this curve is at once interesting to men of 
sciattat aod to practical mechanics. 

15. If iIm generating circle, instead of rolling along a straight line 
is made to ndl upon the circumference of another circle, the curve 
described by aiij point in its circumference is called an epicycloid, 

16. This curve teives.its importance in practical mechanics, from 
its being the curve m«i| frequently adopted for the teeth of wheels. 

17. If the generaduf circle revolves upon the convex circnm- 
ferenee of the quiescent cMe, as in fig. 157, it is then called an 
eMeriar epie^fdoid A B CD. B«t if it revolves on the concave circum- 
ference, as in ^g, 158, the curve h called an interior epicycloid. 

18. The area of an epicycloid, either interior or exterior, may be 
found from the following proportion: as AB:3AB-hBC ::a6c: 
abD i: the area of the generating circle : the area of the epicycloid 
J>BEebD, 

19. If the diameter of the generating circle is equal to half that 
of the quiescent circle, the epicycloid becomes a straight line; which 
circmnstance has been taken advantage of for converting a continued 
circular motion into an alternating rectilinear motion, in the manner 
explained in the article on ^* Select Mechanical Expedients," in the 
Appendix* 



Sbct. V. The Quadratrix. 

The quadrcUrix is a species of curve by means of which the quad- 
rature of the circle and other curves is determined mechanically. 
For the quadrature of the circle, curves of this class were invented 
by Dinoitrates and TechimltauseUy and for that of the hyperbola by 
Mr. Perks. We shall simply describe in this place the quadratrix of 

N 2 



180 curves: catenary. [part i. 

Dinottraies; in order to show its use in the division of an arc or 
angle. 

To construct this quadratrix, divide the quadrantal arc AB (fig. 159) 
into any number of equal parts, A&, hc^ cd^ dB; and the radius AC 
into the same number of equal parts, A/^ /y, gh^ AC. Draw radii 
C^ Cc, &c., to the points of division upon the arc; and let lines 
/li pm, &c.y drawn perpendicularly to AC from the several points of 
division upon it, meet the radii in /, m, n, &c., respectively. The 
curve A/mnD that passes through the points of intersection I, fii,&c. 
is the quadratrix of Dinostrates. 

The figure A CD n ml A thus constructed may be cut out from a 
thin plate of brass, horn, or pasteboard, and employed in the division 
of a circular arc. 

Thus, suppose the arc IL (fig. 160) or the angle IKListobe 
divided into five equal parts. Apply the side AB of the quadra- 
trix upon I K, the point B corresponding with the angle K. Draw 
a line along the curve AS, cutting RL in F. Remove the instru- 
ment, and from F let fall the perpendicular FE upon IK. Divide 
EI into five equal parts, by Prob. VII. page 113, and through the 
points of division draw CM, DN, &c. parallel to £F. Then, through 
their intersections M, N, O, P, draw the lines KM, KN, KO, KP, 
radiating from K, and they will divide the angle IKL into five equal 
parts, as required. 

Note 1 . — If, instead of dividing the arc into equal parts, it were 
proposed to divide it into a certain number of parts having given 
ratios to each other; it would only be necessary to divide EI into 
parts having the given ratio, and proceed in other respects as above. 

Note 2. — If the arc or angle to be divided exceed 90 degrees, 
bisect it, divide that bisected arc or angle into the proposed number 
of parts, and take two of them for one of the required divittona 
of the whole arc. 



Sect. VI. The Catenary. 

The catenary is a mechanical curve, being that which is assumed ^fl 
by a chain or cord of uniform substance and texture, when it is hung"^ 
upon two points or pins of suspension (whether those points be-^^ 
in a horizontal plane or not), and left to adjust itself in equilibrio— * 
in a vertical plane. 

The catenarian curve was for a long while regarded as the proper 
form for the chains of a suspension bridge, but a slight consideration, 
of the above definition will show its inapplicability. For, since the^ 
chains of a suspension bridge ought never to be of uniform substance^ 
and have also in addition to their own weight to support that of the 
roadway, which latter is very differently distributed to the former; 
the form of the curve which the chains will assume is materially 
modified, and approaches more nearly to that of a parabola*. 

• See page 207* 



CHAP. VII.] CUBVBS: CATENARY. 181 

Let AB (fig. 161) be the points of suspension of such a cord, 
AaC^B the cord itself when hanging at rest in a vertical position. 
Then the two equal and symmetrical portions AaCy CbB, both ex- 
posed to the force of gravity upon every particle, balance each other 
predaely at C. And, if one half, as C&B, were taken away, the 
other half, AaC, would immediately adjust itself in the vertical 
position under the point A were it not prevented. Suppose it to be 
prevented by a force acting horizontally at C, and equal to the weight 
of a portion of the cord or chain equal in length to CM; then is C M 
the measure of the tension at the vertex of the curve; it is also 
regarded as the parameter of the catenary. Whether the portion 
AaC hang from A, or a shorter portion, as aC, hang from a, the 
tension at C is evidently the same : for, in the latter case, the resist- 
ance of the pin at a, accomplishes the same as the tension of the line 
at a when the whole AaC hangs from A*. 

Let the line C M which measures the tension at the vertex = />, 
let Cd (or the deflexion as it is termed) ^=Xy ad (or the semi-span) 
= y, Ca=4r, CD = A, AB^d, CaA = C&B = i/. Then, 



l.^=^(hyp.l«g'-±-*-±^!£l±^) 

= ^(hyp. log ^±^) = ^(hyp. log t±£) 

te/>M.logten(45*» -f \%); 

where M == 2*30255851, the number by which the common loe- 
■rithnis must be multiplied to obtain the hyperbolic logarithms f; 
and 8 = the angle which the tangent to the curve at the point a 
makes with the horizon. 

2. If the angle 8 of suspension made between the tangent to the 
curve at A or B, and the horizon be 45°; then d : I :: 1 : 11346. 

3. When /= 2rf, then h = 7966 d, and 8 = TT** 3'. 

4. When the angle S of suspension is 56"^ 2B\ then />, a^ y, and 
4r, are as 1, 0*81, 1-1995, and 1*5089 respectively. In this case f, 
the tension at the point of suspension, is a minimum with re- 
elect to jr. 

5. Generally, y = — tan SM . log tan (i 8). 

Or, logy = log tan 8 -f log {log cot (i 8) — 10} + '^22157 - 10. 
This last formula serving to compute an approximative result. 

• ThiP may easily be determined experimenully, by letting the cofd hung 
very freely over a pulley at C, and lengthening or shortening the portion there 
suspended, until it keqM AaC, in its due position ; then is the portion so hang- 
ing beyond the pulley eqnnl in length to C M. 

t ^ce page 93. 



182 curves: catrnaby. [pabt l 

6. The distance of the centre of gravity of the whole cane 9#, 
from the vertex = J (a? + ^-^ — />)• 

7. The radius of curvature — ^= — : this at the ^eitoxii 

rad. curv. = p, 

8. When S and p are given ; then 

jr = j9 . tan 8 
t s j9 . sec 8 

« = p . sec (8 - I) = ^^ , — 

cos s 

y = J9M . log tan (45° + i 8), 

9. When 8 and z^ or ^ /^ are given s then 

/? = ;3r . cot 8 

^ = JBT . cosec 8 

X ^ z . oosec 8 . versin 8 

y = M4f . cot8 . log tan (45° + ^8). 

10. When 8 and y are given : then 

y 

^ M . log tan (45° -f i 8) 



M . cos 8 . log tan (45° + i 8) 

y . tanS 

^ '^ M . log tan (45° -f ^8) 

y . versin 8 

*^ l( . cos 8 . log tan (45° + i s) 
11. When X and y are given ; then 

log tan (45° -f j 8) _ y ^ 
sec 8 . versin 8 VLx* 

^rom which 8 may he found hy an approximative process ^ alac^ 
X _ ^ _ « . sin 8 



sec 8 — 1 ' " versin 8 ' versin 8 

in all these ca^es t is determined in kn^ of chain or cord of wh^ 
the catenary is actually constructed. 

12. To draw the caUnary mechanicaUy, — If the distance 10 
(fig. 161) hetween the points of suspension, and the depth DCof t^ 
lowermost point, he given, hang one extremity of a fine imifa''^ 
chain or cord at one of the points A, and (letting the chain or e^ 
adjust itself as a festoon in a vertical plane) lengthen or shorten^ 
as it is held near the other end, over a pin at B, until^ when at r^ 



CHAP. VII.] CUBVBfi: CATBHAHr. 183 

it just reacbes the point C : so shall the cord form the catenary ; and 
a pencil passed along the cord, from A by a, C, &, to B, will mark the 
curve upon a vertical board brought into contact with it. 

13. All catenaries that make equal angles with their ordinates at 
their points of suspension are similar, and have j; to y a constant 
ratio : and of any two which do not make equal angles, but have 
X to y in different ratios, a portion may be cut from one curve similar 
to the other. Thus, let ACB and A'C'B' (fig. 162), be the two 
curves, of which A' C" B' is the flatter. Suppose them placed upon 
one axis D C C\ and the tangent T' S', to the lower curve, at B', the 
point of suspension, to be drawn. Then, parallel to T'S' draw 
another line TS to touch the other curve in b. Through b draw 
ba parallel to B^A^. So shall the portion aCb of the upper catenary 
be similar to the lower catenary A'C'B\ 

14. With reference to the practical uses of the catenary, we may 
now blend the geometrical and the mechanical consideration of its 
properties. Taking any portion Cb (fig. 163) of the catenary, from 
the lowest point C; its weight may be regarded as supported by 
tensions acting in the tangential directions C N, by. The strains at 
C and b may be conceived as acting at the point of intersection N ; 
above which, therefore, in the vertical direction N O, the weight of 
the portion Cb may be conceived to act at its centre of gravity, G. 

Hence, strain at C : weight of C^ :: sin ON6 : sin &NC :: cos&NR 
: sin 6 N R :: radius : tan &NR :: radius : tan </^ N. 

Hence, the horizontal tension at C being constant^ the weight, and 
consequently the length of any portion ci of the uniform chain must 
be proportional to the tangent of the inclination of the catenary to 
the horizon at the extremity b of the said portion. This may he 
ttgarded as the characteristic property of the catenary, 

15. In like manner, the horizontal strain at C : oblique strain at 
& :: sin ^ N O : sin C N O :: cos & N R : radius :: radius : sec & N R. 

Therefore, the strain exerted tangentially, at any point 6, is pro- 
portional to the secant of the inclination at that point. 

Also, from § 14 and § 15, tangential strain at b : weight of B & C :: 
wecdby : tani^N. 

These properties evidently accord with the preceding equations. 

16. Let, then, CO, in the axis produced downwards, be eqnal to 
the parameter, or the measure of tne horizontal strain at C ; and upon 
O as a centre with tadius CO dcscHbe a circle. A tangent dt drawn 
to this circle from cf, will be parallel to the tangent &NA of the 
curve at the point b to which db ib the ordinate. That tangent dt 
(to the circle} will also be equal in lencth to the corresponding por- 
tion 5 C of die curve : while the tension at b will be expressed by 
a length of the chain equal to the secant O d. So again, if D T be 
a tangent to the circle drawn from D, it will be equal in Icneth to 
B5c, and parallel to the tangent to the catenary at B; while the 
secant OD will measure the oblique tension at B; evidently, ex- 
eeeding the constant horizontal tension or strain at C, by the 
abadna CB. 



184 CUBVBS: CATENAHY. [pART I. 

17. When the parameter of the catenary, or the line which mea- 
sures the tension at the lowest point, is equal to the deflexion DC; 
if each of these he supposed equal to 1, then A B = 12*6339, the 
length of chain A C B = 3*4641 ; the strain at the points of suspen- 
sion A and B will each he 2, that at the lowest point heing 1 ; and 
the chain at A and B will make an angle of 60^ with the horizon. 

18. If the strain at C he equal to the weight of the chain, and each 
denoted by 1: then AB = *96242, DC = -1180340, the tension 
at A or B = 1*118, the angle of suspension at those points nearly 
26° 34'; the width of the curve is 8*1536 times, and the length 
8*4719 times, the deflexion DC. 

19. If the strain or tension at the lowest point be doable the 
weight of the chain : then if the parameter be 1, A B will be '49493, 
C D = 03078, the strain at A or B 1*03078, the angle of suspenaon 
about 14° 2', the width or span 16*0816 times, and the length of 
chain 16*2462 times the deflexion. 

The magnitudes of the lines, angles, and strains in many other 
cases, may be seen in the table below. The whole theory may be 
yerifled experimentally, by means of spring steelyards applied to a 
chain of given length and weight, placed in various positions, accord^ 
ing to the method suggested at page 244, when trc»Etting of the me- 
chanical powers. 

20. Taking A B = </, C D = A, length A C B = /!» strain at C or 
parameter = p, then, in all cases where the deflexion is small com- 
pared with the length of the chain. Professor Leslie shows*, that 

d' d- 

p = - , -f ] A .... strain atAorB = --r-flA 

or /? = — — I ^ .... strain atAorB=-7 + JA 

^ = ^■^33- 

In this case, too, the strains at G and A or B are nearly in the 
inverse ratio of the deflexion t. 

21. The following table is abridged from a very extensive one 
given by Mr. WarCy in his " Tracts on Vaults and Bridges." Two 
examples will serve to illustrate its use. 

Ex. 1. Suppose that the span of a proposed suspension bridge is 
to be 560 feet, and the deflexion in the middle 25^ feet; what will 
be the length of the chain, the angle of suspension at the extemities, 
and the ratio of the horizontal pressure at the lowest point, and the 
oblique pressures at the points of suspension, with the entire weight 
of the chain ? 

* Elements of Natural Philosophy, p. 63. 

f For a very complete investigation uf the proper forms of catenaries for tns- 
pension bridges, with remarks on the Menai Bridge, and on the failure of the 
suspension bridge at Broughton, see Mr. Eaton Hodgkinaon*s paper in the 
Memoirs o( the Manchester Society, vol. v., New Series. 



CHAP. 



VII.] 



CURVXS: CATBNARY. 



18^ 



TaUe ofBdaiions of Catenarian Curves^ the Parameter being de^ 

notedly 1. 



wiSS^ 


Dtfloioii. 
or DC. 


8«»^.or 


Laofth ot 
chain, or CbD. 


TAMkmat 
the point at 


Semi-ipan di. 
Tided "by the 
deflexion, or 
DB fiC. 


!• C 


•00015 


•01745 


-01745 


1-0001 


114-586 


2 


•00061 


•08491 


-03492 


10006 


67-279 


8 


•00187 


•05288 


•05241 


1-0014 


88-171 


4 


•00244 


•06987 


•06993 


10024 


28-613 


5 


•00882 


•08788 


•08749 


1-0088 


22-874 


6 


•00551 


•10491 


•10510 


1'0055 


19046 


7 


•00751 


•12248 


•12278 


1'0075 


16-309 


8 


•00988 


•14008 


•14054 


1*0098 


14'254 


9 


•01247 


•15778 


-15838 


10125 


12-654 


10 


'01548 


•17542 


-17683 


1'0154 


11-872 


11 


•01872 


-19818 


•19438 


10187 


10-820 


12 


•02284 


-21099 


•21256 


1-0223 


9-444 


18 


•02680 


•22887 


•28087 


10263 


8-701 


14 


-03061 


•24681 


-24983 


1-0806 


8-062 


15 


•08528 


•26484 


•26795 


1-0853 


7-608 


16 


•04080 


•28296 


-28675 


10408 


7-021 


17 


-04569 


•80116 


•80573 


• 10457 


6-691 


18 


•05146 


•81946 


-32492 


10515 


6-208 


19 


•05762 


•88786 


-34433 


1*0576 


5-868 


20 


•06418 


•85687 


-36897 


1'0642 


5-563 


21 


•07114 


•87502 


•38386 


1-0711 


5-271 


22 


•07858 


•89876 


-40408 


1-0786 


5-014 


28 


•08686 


•41267 


-42447 


1-0864 


4-778 


24 


•09484 


•48169 


•44528 


1^0946 


4-562 


26 


•10888 


•45087 


•46681 


11084 


4-361 


28 


•11260 


•47021 


•48778 


1'1126 


4-176 


28 


•18257 


•50940 


•58171 


1-1826 


8-843 


80 


•15470 


•54980 


•57785 


ia547 


8-561 


82 4 


•18004 


-59120 


•62649 


1-1800 


8-284 


84 16 


•21003 


•68710 


•68180 


1'2100 


8-034 


86 52 


•24995 


•69820 


•74991 


1-2499 


2-773 


89 11 


•29011 


•74480 


•81510 


1-2901 


2-567 


41 44 


•84004 


•80290 


-89201 


1-8400 


2-362 


44 


•89016 


•85660 


•96569 


1'3902 


2-196 


46 1 


•48999 


•90660 


1*03610 


1-4400 


2-060 


48 11 


•49981 


•96230 


1-11780 


1-4998 


1-925 


50 8 


•56005 


1-01420 


1-19740 


1-5800 


1-811 


52 9 


•62978 


1-07060 


1-28690 


1-6297 


1-699 


54 18 


•71021 


1-18040 


1-88740 


1-7102 


1-592 


56 28 


•81021 


119950 


1-50890 


1-8102 


1-481 


58 8 


•88972 


1-25100 


1-60340 


1-8897 


1-416 


60 


1^00000 


1-81690 


1-78210 


2-0000 


1-817 


64 6 


1-28940 


1-47020 


2-05940 


2-2894 


1-140 


67 28 


1-60950 


1-61350 


2-41020 


2-6095 


1-002 


67 82 


161680 


1-61640 

1 


2-41820 


2-6168 


0-9998 



186 curves: catenary. [part i. 

Here D B — D C = -280 -h "25 875 = 10'8*2, a namber which is 
to be foand in the table. 

Opposite to that namber^ we find 11° for the angle of saspension, 
DB= 19:318, CB= 19488, tension at A or B = 10187, the 
constant tension at the Tertex being 1. (Fig. 161). 

CoDseqnentlj, 19318 : 19438 :: 560 : 563*48 length of the chain. 
Also, horizon, pressure at C being taken as 1*0000 
the oblique pressure at A or B will = 1*0187 
and the entire weight of chain will = '39876 

Ex. 2. Sappoae that while the span remains 560, the deflexion is 
increased to 51. 

Here D B -f- D C = "284) -;- 51 = 5-49. This number is not to he 
found exactly in the table. The nearest is 5*553 in the last column, 
agreeing with *20^, the angle of suspension. 

Now, 5*55 — 5*49 = 06, and 5*55 — 5'27 = '28, the former 
differenee being oearlj one«>fifUi of the latter. Hence, adding to 
each number, in the line agreeing \iith '20°, one-fifth of the differ- 
ence between that and the corresponding number in the next line, 
we shall have 

An^e of suspension = aO"" 1*2', DC = 06556, DB = -86010, 
CD = -36797, tension at A = 1*10656. 

Hence -36010 : 36797 :: 560 : 572 24, length of chain. 
Also, horixontal pressure at C being taken as 1*0000 
the oblique prest^ure at A or B will ss 1*10656 

and the entire weight of chain will = -73594 

Comparing this Tilth the former case, it n-ill be seen that the 
tensions at C and A, in reference to the weight of the chain, arc 
diminished nearly in the inrerse ratio of the two values of DC; 
thus confirming the remark in art. '20. 

In making use of this table, the remark at page 1 80 must be borne 
in mind, since the results obtained from it, will be only approxima- 
tions more or less correct, as the weight of the chain exceeds that of 
the roadway and load. Where greater accuracy is required, the 
formula given at page 208 must be employed in its stead. 



PT. II. CH. I.] mechanics: 8TATIC8. 18? 



PART II. 

MIXED MATHEMATICS. 



CHAP. I. 
MIOHANIOS IN OBNBRAL. 

1. Meehanies is ^e science of equilibrium and of motion. 

2. Every cause which tends to move a body, or to stop it when in 
motion, or to change the direction of its motion, is callea a force or 



3. The dkreetion of a foree is that straight line in which the point 
to which the force is applied tends to move by virtue of that force. 

4. When the forces that act upon a body, destroy or annihilate 
each other s operation, so that the body remains quiescent, they are 
«id to be in tquiUhrium^ and are then called presntres, 

5. The dirmticn of a pressure is the straight line in which that 
presaore tends to prevent the motion of the point to which it is ap- 
pKad. 

6. The effect of anv force or pressure is found to be the same, at 
wfaattfver point in the line of its wection it be applied. 

7. Two forces or pressures are said to be tqualy when, being ap- 
plied in appoiite directions to the same point, no motion ensues. If, 
however, both forces are supposed to act in the iame direction, the 
single force or pressure which would be required in the opposite 
direction to keep the point in equilibrium, is said to be double either 
of the former forces. And one of the former forces being taken as 
the unit, the latter force would be represented by two of such units. 

8. When two forces can be expressed in terms of the same 
unit, the^ are said to be eommensurcAle. 

9. It IS usual to represent forces and pressures by lines, the diree- 
iioH of the line coinciding with the actual direction of the force, and 
(a line of a certain length being taken as the representative of a unit 
of force,) the ien^ of the line expressing the amount or magnitude 
of the given force or pressure ; and a force thus denoted is said to be 
represented both in magnitude and direction, 

10. When several forces or pressures act in different directions 
upon the same point, it is possible to find the direction and magnitude 
of another force or pressure which would replace all the others ; that 



188 mechanics: statics. [pabt ii. 

is, if the others were remored, and this one force applied in their 
stead, the effect produced would he the same. Sach a force is called 
the resultant of the others, which are named the components^ and the 
operation which we have supposed is called the composition of forces, 
and its converse, or finding the directions and magnitudes of any 
nnmher of forces which would produce the same effect as any one 
given force, is called the resolution of forces. 

11. The moment of a force or pressure ahont any point, is the 
perpendicular distance from that point to the line of direction in 
which the force acts, multiplied hy the numher of units expressing 
that force. 

12. Vis inertia^ or power of inactivity, is defined hy Newton to 
be a power implanted in all matter, by which it resists any change 
attempted to be made in its state, that is, hy which it requires force 
to alter its state, either of rest or motion. 

13. Vis vioiiy or living force, a term used by Leibnitz to denote 
the force or power of a body in motion; or the force which would 
be required to bring it to a state of rest. 

14. Mechanics is usually divided into five branches; viz. — 

I. Statics^ which relates to the equilibrium of pressures applied 
to solid bodies, and of the weight and pressure of bodies when 
at rest. 

II. Dynamics^ which relates to tlie motion prodaced in solid 
bodies by the application of force. 

III. ffydrostaticsy which relates to the equilibrium and pressure 
of non-elastic fluids, and of the weight, pressure, and sta- 
bili^ of bodies immersed in them. 

IV. NydrodynamicSy which relates to the motion of non'-elaaie 
fluids by the application of force. 

V. Pneumatics^ which relates to the various circumstances at- 
tending the equilibrium or motion of elastic fluids. 



CHAP. II.] STATICS. 189 



CHAP. II. 

STATICS. 
Sect. I. Statical Equilibrium, 

1. Ip any two pressures applied to a point keep it in eqailibrinro, 
they most be eqnal to each other, and must act in the same straight 
line and in contrary directions. 

3. If any tkree pressures applied to a point keep it in equilibrium, 
they must all act in the same plane ; and any one of those pressures 
is represented in magnitude and direction by the diagonal of a paral- 
Ukaram^ whose sides represent the other two pressures in magnitude 
and direction. 

3. If nnj/our pressures whose directions are not in the same plane, 
applied to a point, keep it in equilibrium, any one of those pressures 
is represented in magnitude and direction by the diagonal of a paral^ 
lelapipedon^ whose contiguous edges represent the other three pres- 
sures in magnitude and direction. 

4. If any number of pressures whose directions are all in the 
same plane, applied to a point, keep it in equilibrium, those pressures 
will be represented in magnitude and direction by a polygon whose 
M&i are made parallel to the direction of those pressures, and pro- 
portional to their magnitude. 

5. If any number of pressures in the same plane be in equili- 
brium, and any point be taken in that plane from which their mo- 
ments are measured, then the sum of the moments of those pressures 
which tend to turn the plane in one direction about that point, will 
be equal to the sum of the moments of those which tend to turn it 
in the opposite direction. 

6. If any number of pressures acting in the same plane, and which 
are in equilibrium about a given point, be moved parallel to the direc- 
tions in which they act, until they all intersect in any given point, 
they will still be in equilibrium about that point. 

7. The resultant of two parallel pressures, if acting in the same 
direction, is equal to their 8um; but, if acting in contrary directions, 
is equal to their difference^ the direction of their resultant being in 
the direction of the ereater pressure. And generally the resultant of 
any number of parallel pressures is equal to the several pressures 
added together with their proper signs. 



190 STATICS. [part II. 

From the foregoing propositions we may deduce the following for- 
mulffi: — 

8. Let P. and P^ represent as j two pressures, and ff the angle 
formed hy tneir two lines of direction ; kt R be their resultant ; y 
the angle which its line of direction makes with that of Pj, and ^ that 
which it makes with P^; then 

R = >/Pi* + Pj' + 2 Pj Pj cos /9 . . . (I.) 

P.. sin $ ,-- . 

tan y = - — ~- ^ (II.) 

'^ Pi + P« cosg ^ ^ 

Pj : Pg :: sin ^ : sin y (IH.) 

or, three forces being in equilibrium, any two of them are to each 
other inversely as the sines of their inclinations to the third. 

9. To determine the resultant of any number of pressures in the 
same plane: — Let P,, P^, P,, &c. represent the pressures in magni- 
tude, and a J, a^, a 3, the inclinations which their lines of direction 
make with some line given in position ; let R be their resultant, and 
» its inclination to the same line; then 

R cos » = P| cos a, -f- Pg cos aj -f .... + P« COS «« . (IV.) 

R sin » = Pj sin a, + P^ sin a^ + . . . . + P« sin «, . . (V.) 

in which the several terms are taken positive or negative according 
to the direction in which they act. From these we obtain 

R = {>/(P, sinai+Pjsina, + + P,sin«,)* + 

(Pi cos a, + Pg cos «^ -f + P, cos aj«} . . . (VL) 

t^„,^P|Bina, 4-P.8ina,.f.... + P,«i°a, ...(VU.) 

P, cos a J 4- Pg COS«j 4- . . . . -h P^ COS «, 

10. Let Pj, Pg, P., &c., be any number of parallel pressures, 
(being + or — according to their direction,) and D,, D^, D^, Ac, 
be their perpendicular distance from any given parallel plane : let B 
be their resultant, and A its perpendicular distance from the same 
plane: then 

R = P, + P2 -f + P (VIII.) 

^^P»I>>-i-P,D,+ 4-P,D, 

Pi + P« + . . . -h P. ^ ' 

As these formulae and propositions are of universal application in 
the constructions of civil engineers, architects, and mechaniciaos, we 
shall give a few simple examples, to render tlieir use clear to those 
who are not familiar with mathematical form'ulee. 

Ex, 1. Suppose that a weight B is attached by a stirrup to the foot . 
of a king-post A B, which is attached to two rafters A C, A D, in the 
respective positions shown in fig. 1 64. Then if A E be set off upon 



CHAP. II.] STATICS. 191 

AB, equal in numerical value to the yertical strain upon A B, and the 
parallelogram AFEO be completed, AF measured upon the same 
scale will show the strain upon the rafter A C, and A O the strain 
upon A D. 

Ex. 2. Let it be proposed to compare the strains upon the tie- 
beams AD, and the struts AC, when they sustain equal weights B, in 
the two different positions indicated by figs. 165 and 166. Let A £ in 
one figure be equal to the corresponding yertical line A£ in the 
other, and in each represent the numerical value of the weight B, 
that hangs from A. Through E in both figures, draw lines parallel 
to D A, A C, respectively, and let them meet A C, and D A produced 
in F and O: then AFEG in each figure is the parallelogram qf/orces 
by which the several strains are to be measured. A G represents the 
tension upon the tie-beam A D, and A F the strain upon the strut 
A C. Both these lines are evidently shorter in the lower figure than 
they are in the upper, A E being of the same length in both : there- 
fore the first figure exhibits the least advantageous position of the 
beams. It is evident also, that while C A tends upwards and D A 
downwards, the greater the angle D A C, the less is its supplement 
C A G, and the less the sides F A, A G, of the parallelogram. 

JSm. 3. Let it be required to determine geometrically, and by com- 
potadon, the magnitude and direction of the four pressures P^, P^, 
Pj and P^, (fie. 167,) all applied to the point A, and acting in the 
sme plane; let Pj = 12, P^ = 9, P, = 16, P^ = 15, and the 
angles which their directions make with the line BAG, (given in 
pontion,) be 77^ 37% 9° and 48% respectively. 

1. CrecmeiricaUy. — From the point A (fig. 168) lay off the angle 
CAD equal 77% and upon the line A D set off 12, from any scale of 
equal parts ; then through D draw the line D E, equal to 9 parts by 
the same scale, and making an angle of 37° with B A C ; again, 
through £ draw the line £ F equal to 16, and inclined 9° to B A C ; 
and lastly, through F draw £ G, equal to 15, and inclined 48° to B A C; 
then the straight line AG, connecting the first and last points, and 
completing the polygon A D £ F G A, will be the resultant of the four 
given pressures ; and being measured by the same scale, will be found 
equal to 36*72 parts nearly, and to be inclined 13° to B A C. 

2. By Computation.— k^ the pressures P^, Pg) and P., are not on 
the same side of the line BAC (fig. 167) as P^, the signs of their 
ftfies most be taken differently ; but as they are ail on the same side 
of the line H A I, (perpendicular to B A C,) the signs of their cosines 
must be taken alike; then if R is their resultant, and » its inclination 
to B A C, we obtain from formula (VI.) — 

E=^/{(ia X -9744 + 9 X -6018+ 16 X 1664-15 x •7431)' -f 

(12 X -225 + 9 X -7986 -f 16 x -9877 + 16 x -6691)*} = 36-7138; 

•ad from formula (VII.) — 

12 X -9744 + 9 X -6018 + 16 x 1564 - 15 x '7431 ^^,,,^ 

ttn«B3 . ='23106 

12 X -225 + 9 X -7986 + 16 x -9877 + 13 x 6691 

.-. • = 13^ 

In which the values are the same as were found geometrically. 



192 CENTRE OF GRAVITY. [PABT II. 



Sect. II. Centre of Gravity, 

1. Gravity is the force in virtue of which bodies left to themselves 
fall to the earth in directions perpendicular to its surface. 

2. We may distinguish between the effect of gravity and that of 
weighty by observing that the former is the power of transmitting to 
every particle of matter a certain velocity which is absolutely inde- 
pendent of the number of material particles ; while the latter is the 
effort which must be exercised to prevent a given mass from obeying 
the law of gravity. Weighty therefore, depends upon the fiuu$; but 
gravity has no dependence at all upon it. 

3. The centre of gravity of any body, or system of bodies, is that 
point about which the body or system, acted upon only by the force 
of gravity, will balance itself in all positions: or it is a point which, 
when supported, the body or system will be supported, however it 
may be situated in other respects. 

The centre of gravity of a body is not always within the body itself; 
thus the centre of gravity of a ring is not in the substance of the 
ring, but in the axis of its circumscribing cylinder; and the centre of 
gravity of a hollow staff, or of a bone, is not in the matter of which 
it is constituted, but somewhere in its imaginary axis; every body, 
however, has a centre of gravity, and so has every system of bodies. 

4. Varying the position of the body will not cause any change in 
the relative position of the centre of gravity ; since any such muta- 
tion will be nothing more than changing the directions of the forces, 
without their ceasing to be parallel ; and although the amount of the 
forces may not continue the same, in consequence of the body bein? 
supposed at different distances from the earth, still the forces upon aO 
the molecules vary proportionally, and the position of their centre re- 
mains unchanged. 

5. When a heavy body is suspended by any other point than its 
centre of gravity, it will not rest unless that centre is in the same 
vertical line with the point of suspension ; for in all other positions 
the force which is intended to ensure the equilibrium will not be di- 
rectly opposite to the resultant of the parallel forces of gravity upon 
the several particles of the body, and of course the equilibrium will 
not be obtained. (See Art. 9, on Pendulums^ page 222.) 

6. If a heavy body be sustained by two or more forces, their 
directions must meet, either at the centre of gravity of that body, or 
in the vertical line which passes through it. 

7. When a body stands upon a plane, if a vertical line passing 
through the centre of gravity fall within the base on which the body 
stands, it will not fall over; but if that vertical line passes without the 
ba^e, the body will fall, unless it be prevented by external means. 
When the vertical line falls upon the extremity of the base, the body 
may stand, but its state (which is called unstable equilihriuni) may 
be disturbed by a very trifling force ; and the nearer this line passes 
to any edge of the base the more ensily may the body be thrown 




CHAP. II.] CBNTRB OF GRAVITY. 193 

OTer ; the nearer it falls to the middle of the hase, the more firmly 
the body stands. 

Upon this principle it is that leaning lowers have been bnilt at Pisa, 
and various other places ; the Tertical line of direction from thecentre 
of gravity falling within the 
base. And, from the same 
principle it may be seen , ^^^ 
that a waggon loaded with ^fl^|p^|^ 
heavy materials, as B, may 
stand with perfect safety, on 
the side of a convex road, 
the vertical line from the 
centre of gravity falling be- 
tween the wheels ; while a waggon A with a high load, as of hay, or 
of wool-packs, shall fall over, because the vertical line of direction 
fells without the wheels. 

8. Owing to the great distance of the earth's centre from its sar- 
&ce, the directions of the force of gravity of the several molecule 
composing a body may be considered parallel without any appreciable 
error, and therefore all that has been said in the foregoing section on 
the subject of parallel pressures will apply to the force of gravity ; 
and the formula for finding the position of the resultant of any num- 
ber of pandlel pressures, may be applied to determine the position of 
the centre of gravity. 

Thas, if Bj, B^, B3, &c., denote the weights of the particles of 
any body, and D^, D„, D,, &c., the perpendicular distances of their 
reflective centres of gravity from any given plane ; then, the dis- 
tance TA) of their common centre of gravity from the same plane is 
found by formulsB (IX.) to be 

^ ^ B^D, +B^ Dg -I- +B,D, 

Bj +B2 + +B. 

Therefore, if by means of this formula we ascertain the distance of 
the centre of gravity from any three planes given in position, we shall 
have determined its exact situation in space. 

9. The common centre of gravity of two bodies divides the right 
line which joins their respective centres of gravity, in the inverse 
ratio of their weights. The centre of gravity of any number of bodies 
may be found, by finding the common centre of gravity of any two of 
the bodies, and then considering this as the centre of gravity of a 
body equal to the combined weight of the two, we may find the com- 
mon centre of gravity of this imaginary body and a third ; and thus 
proceed, ad libitum, 

10. If the particles or bodies of any system be moving uniformly 
and rectilineally, with any velocities and directions whatever, the 
centre of gravity is either at rest, or moves uniformly in a right line. 

Hence, if a rotatory motion be given to a body and it be then left 
to move freely, the axis of rotation will pass through the centre of 





194 CIKTRB OF ORAYITT. [PABT H. 

grwnty : for that centre, either remaining at rest or moving nniformly 
forward in a right line, has no rotation. 

Here too it may be remarked, that a forte appUed at the centre of 
^ravit^ of a body^ eannct produce a rotaUny motion. 

11. To find the centre of gravity meebanically, it is only requisite 
to dispose the body successiYely, in two positions of eqoilibriam, by 
the aid of two forces in vertical directions, applied in succession to 
two differMt points of the body; the point of intersectkm of these 
two directions will show the centre. 

This may be exemplified by particularising a few methods. If the 
body have plane sides, as a piece of board, hang it up by any point 
(A fig. 169), then a piuiAb-line suspended from the same peine will 
pass through the centre of gravity; therefore toark that line (AB) 
upon it : and after suspending the body by another point (C), apply the 
plummet to find another such line (C D) ; then will th^ interaeetion 
show the centre of gravity. 

Or thus : hang the body from a tack successively by two strings 
attached to different parts of it, and each time mark upon it the^line of 
a plummet attached to the same tack ; then will the intersection of the 
two lines be the centre of gravity. 

Another method : Lay the body on the edge of a triangular prism, 
or such like, moving it to and fro till the parts on both sides are in 
equilibrio, and mark a line upon it close by the edge of the prism: 
balance it again in another position, and mark the fresh line by the 
edge of the prism; the vertical line passing through the mterseetion of 
these lines, vrill likewise pass through ^e centre of gravity. The 
same thing may be effected by laying the body on a table, till it ie 
just ready to fall off, and then marking a hue upon it by the edge of 
the table : this done in two positions of the boay will in like manner 
point out the position of the centre of gravity. 

Wlien it is proposed to find the centre of gravity of the arch of a 
bridge, or any other structure, let it be laid down accurately to aoale 
upon pasteboard ; and the figure being carefully cut out, its centre of 
gravity may be ascertained by the preceding process. 

12. The centre of gravity of a right line, or of a paraUdlogram, 
prism, or cylinder, is in its middle point; as is also that of a circle, or 
of its circumference, or of a sphere^ or of a regokr polygon ; the 
centre of gravity of an ellipse, a paraboks a cone, a conoid, a sj^ 
roid, &c., is situated in its axis. And the same of all symmetrioal 
figures. 

13. The centre of gmvity of a triangle is the point of intersectioa 
of lines drawn from the three aneles to the middles of the sides re* 
spectively opposite: it divides ea(m of those lines into two portions ia 
tlie ratio of 2 to 1. 

14. In a Trapemum. Divide the figure into two triangles by the 
diagonal AC (fig. 170) and find the centres of grarity £ and F of 
these triangles; join £ F, and find the common centre O of these twe 
by this proportion, ABC : ADC :: FO : EG, or ABCB : ABC :: 
£F : £0. Or, divide the figure into two triangles by a dii^nal 



CHAP. II.] CmRB OP GRAVITY. 195 

BD; then find thm centres of gravity; and the line which joins 
them will intersect £F in O, the centre of grayity of the trapezium. 

15. In like manner, for any other plane fignre, whatever he the 
numher of sides, divide it into several triangles, and find the centre 
of gravity of each ; then connect two centres together, and find their 
common centre as ahove; then connect this and the centre of a third, 
and find the common centre of these; and so on, always connecting 
the last found common centre to another centre, till the whole are 
included in this process; so shall the last common centre be that 
which is required. 

16. The centre of gravity of a circular arc is distant from the 
centre of the arc a fourth proportional to the arc, the radius, and the 
chord of the arc. 

17. In a circular sector, the distance of the centre of gravity from 

2 c r 
the centre of the circle is— — ; where a denotes the arc, e its chord, 
oa 

and r the radius. 

18. The eentres of gravity of the mrfaees of a cylinder, of a cone, 
and of a conic frustum, are respectively at the same distances from 
the origin as are the centres of mvity, of the parallelogram, triangle, 
and trapezoid, which are verticsl sections of the respective solids. 

19. In the segment of a sphere, or spheroid, A being the whole 
ans, and k the height of the segment, the distance of the centre of 

A A ._ dk h 

gravity from the vertex is equal -^--^ 7-7 ; and when the height be- 

o A — 4 A 

oomes half A, or the segment becomes a hemi-sphore, the distance 

Irom the vertex is equal to ^ A. 

20. The centre of grarit^ of the surface of a spheric segment is 
at the middle of its versed sine or height. 

21. The centre of gravity of the convex turface of a spherical 
lone, ia in the middle of that portion of the axis of the sphere which 
is intercepted by the two bases of the zone. 

22. In a cone, as well as any other p3rramid, the distance from the 
vertex is \ of the axis. 

23. In a conic frnstnm, the distance on the axis from the centre 

of the leaser end, is J A. -j- : whore h equals tlie 

he^fat, and R r the radii of the greater and lesser ends. 

24. The same theorem will serve for the frustum of any regular 
pyramid, talcing R and r for the sides of the two ends. 

25. In the paraboloid, the distance from the vertex is f of the 
axis. 

26. In the frustum of the paraboloid, the distance on the axis from 

2 R* + r* 
the centre of the lesser end, is J h. - : where h equals the 

hcigfat, and R r the radii of the greater and lesser ends. 

2 



196 IQUILIBBIUM OF PIBB8. [PABT H. 

27. Every figure, whether superficial or solid, generated bj the 
motion of a line or surface, is equal to the product of the generadng 
magnitude multiplied bj the path of its centre of gravity. 

As an example, let A B C (fig. 171) be a right angled triangle, the 
revolution of which on the leg A B will produce a cone ; let A B =: 9, 
BC = 6, and D be the centre of mvity; then by § 13, AD ^ | AF, 
therefore D£ sfFB = Q equal the radius of the circular path de- 
scribed by the centre of gravity, the circumference of which will 
therefore be 4 x 31416 = 125664, which multiplied by (3 x 9) 
= 27, the area of the triangle, will be 339*2928, the solid content of 
the cone; and by the rule given at page 130 for finding the solidity 
of a cone, we obtain 12* x -7854 x 3 = 3392928, the same 
result. 



Sect. III. General application of the principles of StaUcs to 
the equilibrium of Structures, 

Every structure is exposed to the operation of a system of forces; 
so that the examination of its stability involves the application of the 
general conditions of equilibrium. 

Now, no part of a structure can be dislocated, except it be either 
by a progressive, or a rotatory motion. For either the part is dis- 
placed, without changing its form^ in which case it is (as a system of 
invariable form) incapable of receiving any instantaneous motion, 
which is not either progressive or rotatory ; or else it happens to be 
displaced, changing at Vie same time its form^ which, considering the 
cohesion of its parts, cannot take place without its breaking in its 
weakest section ; in which case a progressive motion is generated if 
the force acts perpendicularly to the section, and a rotatory motion, 
if it acts obliquely. 

We shall here consider the most useful cases ; indicating by the 
word stress^ that force which tends to give motion to the structure, 
and by resistance^ that which tends to hinder it. 

EQUILIBRIUM OF PIERS, OR ABUTMENTS. 

1. If we suppose figure 1 72 to be the vertical section of a pier, we 
may reason upon that section instead of the pier itself, if it be of 
uniform structure. 

Let O be the place of the centre of gravity of the section A B C D, 
S Z the direction in which the stress acts, meeting X I, the verticai 
line through the centre of gravity, in I. Then, supposing the stress 
to be resolved into two forces, one (Pj) vertical, the other (P^) 
horizontal ; the pier (regarding it as one bodv) can only give way 
either by a progressive motion from B towards A, or by a rotatory 
motion about A. 

2. The progressive motion is resisted by friction. If W denote 
the weight of the pier, Pj the stress estimated vertically, and Pj its 
horizontal effort, then the pressure on the base sW + P^, and 



CHAP. II.] EQUILIBRIUM OF PIERS. 197 

potting /for the coefficient of friction, its friction =/(W + Pj, 
which 18 the amount of the resistance to progressive motion. So that 
to ensure stahilitj in this respect we must have 

/(W + PJ> P, (I.) 

8. To ensure stahilitj in regard to rotation, taking the moments of 
the forces ahout the point A, we must have 

W . AX -f Pj . Xe > P, . Tq (II.) 

Or, supposbg the stress not to he resolved, its moment ahout tho 
point A must be less than that of the weieht of the pier about the 
same point ; that is, putting S for the whole stress, 

W.AX>S.aY (III.) 

Or, by a graphical process, suppose the two forces W and 8 to he 
applied at I, and complete the parallelogram, having sides which 
represent these forces. Then must the diagonal representing the 
resultant of those forces produced meet the base on the side of A, 
towards B, to ensure stability. 

4. If, as is very frequently the case, the vertical section of the 
pier is a rectangle, putting k for the height of the pier, h for its base, 
w for the weight of a cube unit of the material of which it is formed, 
and R a horizontal stress applied at the summit of the pier ; then 
the pier will be in a state of unstable equilibrium, as far as regards 
its progressive motion, when 

hfhw^lBi :.... (IV.) 

and as r^grds its being overthrown when 

V=* (V) 

But in order that the structure may really be secure, these equations 
most become 

b/hw>B^ and — ->R, 

and the more the first member exceeds the second the greater will 
be the stability of the pier. 

EwampU 1. Suppose a rectangular wall 89*4 feet high, and of a 
material weighing 125 lbs. per cube foot, is to sustain a horizontal 
atnin of 99(K) lbs. avoirdupois on each foot in length, applied at its 
somiDit: what must be its thickness that there may be an equilibrium, 
taking /= 1*75. 

By transposing formula (IV.) above, we obtain 

and substituting the several values given above, we have 

h = = 2-68 feet; 

•75 X 89-4 X 125 



198 PRESSURI OF EARTH AOAlVWt WALLS. [PABT II. 

therefore the thickness of the pier at its base mast be more than 
2*68 feet, otherwise it will be moved horizontally by the stress R. 
But we must also inquire, what thickness is requisite to prevent the 
pier being overthrown; and this we shall ascertain from formula 
(v.), which by transposition becomes 



V2R 



/2 X9900 __^ 
and * = A / — r^T — = 12*58 feet. 



Therefore, we see, that the pier most be more than 12*58 feet 
upon its base, otherwise it will be overthrown by the horizontal 
strain at R. 

Here, as the thickness required to prevent overturning is much 
the greatest, the computation m reference to the other kind of equili- 
brium may usually be avoided. 

Example 2. An embankment, or dam, A B D (fig. 17d), formed 
of clay between two rubble walls, is exposed to the pressure of 
80 feet of water ; it is required to ascertain whether the dam will be 
overthrown, and also whether it will slip upon its bed. Its dimen- 
sions are, D C = 4 feet, B C which is vertical = 32 feet, and A B 
Bs 12 feet; its weight may be taken at 135 lbs. per cubic foot, and 
the coefficient of friction or f ^ '5. 

First, by the method laid down in § 14, page 194, we ascertain 
the distance of the centre of gravity of the wall from its interior face 
B C, to be 4*333 ; therefore the distance (A X) of its line of direc- 
tion from A will be 7*667 feet. The weight of the wall (W) will be 
12+4 
— - — X 32 X 135 = 34560 lbs.; the pressure of the water 

30 
(Pg or 8) will be -— X 62*5 = 937*5, and the centre of pressure 

of the water being at two-thirds of its depth (as explained at page 250), 
the distance (AZ) of its line of direction above the point A will 
be 10 feet. 

Now, in order to ascertain its stability to resist progressive motion, 
we must substitute these values in formula (I.), when we obtain 
•5 X 34560 > 937*5, or 
17280 > 937-5; 
therefore there would be no fear of the embankment slipping upon 
its base. And from formula (III.) we have, 

34560 X 7-667 > 937-5 X 10, or 
264972 > 9375 ; 
therefore the wall cannot be overthrown. 

PRE88URB OF EARTH AGAINST WALLS. 

1. Let DACB ^fig. 174) be the vertical section of a wall behind 
which is posited a oimk or terrace of earth, of which a prism whose 



CHAP, n.] raammR of xabth against walls. 199 

section is represented by CBH wonid detach itself and fall down, 
were it not preyented by the wall. Then B H is denominated the 
line of rupture or the natural slope^ or natural deeltvity. In saudy 
or loose earth, the angle CHB seldom exceeds 30°; in stronger 
earth it becomes 37°; and in some favourable cases more than 45°. 

2. Now, the prism whose vertioal section is C B H, has a tendency 
to descend along the inclined plane B H by reason of the force of 
gravity ; bat it is retained in its place, not only by the force opposed 
to it fy the wall, bat also by its cohesive attachment to the face B H, 
aDd by its fiicdon upon the same surface. 

If we resolve the weight of this prism into two forces, one acting 
pcfpendicular to the plane B H, and the other parallel to the same, 
the latter minus the force required to overcome either the cohesion 
or the friction of the surface B H, will be the strain acting in a direc- 
tion passing through the centre of gravity of the prism CBH and 
parallel to B H, upon the back of the wall, and tending to overthrow 
It ; and the aiBount of this atraio, and its direction, being ascertained, 
the formulsB already given may be employed to determine the sta- 
InU^of the wall, 

I H 2s LH + eohesioii + friction. 

S. It is evident, therefore, that m the angle at whidi the earth 
will stand U one <>f the elements in the calculation, the solution to 
this inquiry must be, in a great measure, experimental. It has been 
found, however, theoretically, bv M. Prony*, and confirmed experi- 
mentiJly, that the angle formed with the vertical (CBH) by the 
prism ojf earth that exerts the greatest horizontal stress against a 
wall, is half the angle which the natural slope of the earth makes 
with the vertical : and this emrious result greatly amplifies the whole 
inquiry. 

Puttisg A for the heiffht of the wall, ff for the angle CBF, or 
half C B H, the natural slope of the ground, w^ for the weight of a 
cabit onit of the ground, and /for the coefficient of friction we have 

h^w^ . tan/S_ 
2 

the weight of the prism CBF; and resolving this into two forces, 
one (P,) perpendicular, the other (P^) parallel to BH, we have 
for the former 



and for the latter 



J^w^ .imp .m$ ^ 

5 P,, 



2 "" ^ 



* See a denxmstnetion at p. 309, vol. il. tenth edition of Dr. Hatton't 
ConiM of MatbcmiitiGt. 



200 PRBS8URB OP EARTH AGAINST WALLS. [PART 1% ' 

Therefore tlie friction along the surface B H, will be 
fh^w^ . ton /9. sing y, . 

which subtracted from P^, gives 

(l_/.tang)*!i^l^=S (VII.) 

equal the strain acting through the centre of gravity of the prism^i- 
upon the back of the wall ; now, as the centre of gravity of a triangle^ 
is situate at a third of its height, and the direction of the strain S i^ 
parallel to the sloping base B H, it will meet the wall at one-third or" 
its height, and b being put for the breadth of the base of the wall, 
we shall have 

st^.sinp-b.coBp\ (VIII.) 

equal the moment of the strain S about the point A, tending to 
overthrow the wall. 

Example, Suppose a wall is to be built of brickwork (weighing 
117 lbs. per cubic foot), to support a terrace 39 feet in height, the 
earth composing which weighs 105 lbs. per cubic foot, its natural 
slope being 53° from the vertical, and the coefficient of friction *45 ; 
it is required to find the breadth which must be given to the wall at 
its base in order to ensure its not being overthrown. 

By substituting these values in formula (VIIL), we obtain 

S = (1 - -45 X -4986) «^' X iO^ X '^^^^ = „630; 
the moment of which from formula ( VIII.) is 

27636 f^ X -4462 - ^ x 8949^ = 160322 — 24731 . b. 
Therefore by formula (III.), we have 

^^^^ "^ > 160322 - 24731 b, or 

b > 4-58 ; 
that is, the base of the wall must be more than 4*58 feet, in order 
that it shall not be overthrown by the pressure of the earth. 

4. Of the experimentol results, the best which we have seen are 
those of M. Mayniel, from which the following are selected; in all of 
which the upper surface of the earth and of the wall which supports 
it are supposed to be both in one horizon tol plane. 

1st. Both theory and experiment indicate that the resultant of the 
thrust of a bank, behind a vertical wall, is at a distance B K from the 
bottom of the wall equal to one-third of its height. 

2dly. That the friction is half the pressure, in vegetoble earths, 
and four-tenths in sand; or tliaty=s *5 in one case and '4 in the 
other. 



.11.] 



PRBS8UBB OF BABTH AGAINST WALLS. 



201 



ly. The cohesion which vegetahle earths acquire, when cut in 
and well laid, course hy course, diminishes their thrust hy full 
kirdt; or in this case y = '667» 

16 following tahle contains the value of the angle C B F, (heing 
the angle formed hy the natural slope and the vertical,) for 
al different kinds of earth, the authorities heing given in the last 
on. 



Nature of Earth. 



Angle 



Authority. 



Inbble 

Axme shingle perfectly dry 

2arth the most dense and compact 

.^ommon earth tliffhtlv damp . 

Idem pulverized and dry 
Vegetahle earth 

Idem mixed with large g^vel 

Idem mixed with imall g^vel 

Sand 

'Ine dry sand .... 

Idem 

Idem (a single experiment) 



22 30 

26 30 

17 30 

18 
21 35 
31 43 
31 43 
35 52 
34 6 
25 30 

27 45 
34 30 



MaynieL 
Pasley. 
Barlow. 
Rondelet. 

Id. 
MaynieL 

Id. 

Id. 

Id. 
Barlow. 
Rondelet. 
Oadroy. 



THICKNESS OF WALLS, BOTH PACES VERTICAL. 

The following tahle exhibits the thickness which ought to he 
i to a parallel wall of various materials, and supporting the 
ore of different kinds of earth. 





Thickness of the wall or DC, 
its height being 1. 


Nature of Earth. 


WaUof 

brick. 

100 Ibt. 

pereub. ft. 


WaU of un- 
hewn itooe, 

lasibs. 
per cab. ft. 


Wall of 

hewn 

flreotooe. 


taUe earth, carefully hud course by course 

, weU rammed 

li mixed with Uu-ge gravel 


•16 

17 
•19 
•33 


•15 
•16 

•17 
•30 


•13 
•14 
•16 
•26 



. Far waUs with an inUricr slope^ or a slope towards the hank, 
16 hose of the slope he — of the height, and let S and « he the 
fie gravities of the wall and of the earth ; then 

DC =^ A/ -— -I- m ; 

V 3»* S n 

« m ^ •0424, for vegetahle or clayey earth, mixed -with large 



202 BQUILIBBIUM OF P0LT€K)1I8. [PABT n. 

graTe] ; *0464>, if the earth be mixed with mall grmTel ; '1528, for 
sand; and '166, for semi-fluid earths. 

Example, Suppose the height of a wall to be 20 feet, and ^ of 
the height for the base of the tcdm9 or slope; suppose, also, the 
specific gravities of the wall and of the bank to be 2600, and 1400, 
and the earth semi-fluid; what, then, must be the thickness of the 
wall at the crown ? 

Here the theorem will become. 



^^='^Vi^+-*^^-^^-^» 



= 20 >/ -0008333 + -0894 — 1 = (20 x '3) - 1 
= 6 — 1 = 5 feet : 
while the thickness of the wall at bottom will be 6 feeL 

EQUILIBRIUM OF POLYGONS. 

1. Let there be any number of lines, bars, or beams, AB, BC, 
CD, DE, &c. (fig. 175), all in the same vertical plane, connected 
together and freely moveable about the joints or angles. A, B, C, D, 
£, &c., and kept in equilibno by weights laid on the angles : it is 
required to assign the proportion of those weights ; as also the force 
or push in the direction of the said lines ; and the horizontal thrust 
at every angle. 

Through any point, as D, draw a vertical line a J} fig; to which, 
from any point, as C, draw lines in the direction of, or parallel to, 
the given lines or beams, viz., C a parallel to A B, Ch parallel to BC, 
Ce to DE, C/to EF, Cy to F O, &c. ; also CH parallel to the 
horizon, or perpendicular to the vertical line adg^ in which also 
all these parallels terminate. 

Then will all these lines be exactly proportional to the forces 
acting or exerted in the directions to which they are parallel, whether 
vertioil, honxontal, or oblique. That is, the oblique forces or thrusts 

in the direction of the bars AB, BC, CD, DE, £ F, F6, 

are proportional to their parallels ... Ca, C6, C</, C^, CJ\ Cg; 
and the vertical weights on the angles B, C, D, E, F, 9ic. 

are as the parts of the vertical a 5, &D, De, ej\ fg^ 

and Uie weight of the whole frame ABCDEFQ, 

is proportional to the sum of all the verticals, or to ag\ 

also the horizontal thrust at every angle, is everywhere the same 
constant quantity, and is expressed by the constant horizontal 
line CH. 

Cord, 1. It is worthy of remark that the lengths of the bars A B, 
B C, &c. do not affect or alter the proportions of any of these loads 
or thrusts ; since all the lines C a, C ^, a 6, &c., remun the same, 
whatever be the lengths of A B, B C, &c. The positions of the bars, 
and the weights on the angles depending mutually on each other, as 
well as the horizontal and oblique thrusts. Thus, if di^re be givea 



CBAP. n.] BQUILIBBIUM OF P0LT001V1. 203 

the positioo of D C, and the weights or loads laid on the angles D, 
C« B; set these on the Tertical, DH, D6, ba^ then C&, Ca, give 
the direetiooB or positions of C B, B A, as well as the quantity or 
proportion of C H the constant horizontal thrnst. 

Vorol. S. If C H be made radios ; then it is evident that Ha is the 
tngent, aad C« the secant of the elevation of C a or A B ahove the 
horiaon ; also H 6 is %he tangent, md C b the secant of the elevation 
of C^ or CB ; also H D and CD the tangent and secant of the eleva- 
tion of C D ; also H e and C s the tangent and secant of the devadon 
of C« or D B ; also B/ and C/ the tangent and secant of the eleva- 
tion of EF; and so on; also the parts of the vertical a&, 6D, e/, /y, 
denoting the weights laid on the several angles, are the differences of 
the said tangents of elevations. Hence then in general, 

1st. The oblique thmsts, in the directions of the bare, are to one 
another, directly in proportion to the secants of their angles with the 
borixon; or, which is the same thing, reciprocally proportional to 
the eosines of the same angles, or reciprocally proportional to the 
sines of the vertical angles, a, ^, D, e, /^ y, &c., made by the vertical 
line with the several directions of the bars; because (formula K . 3, 
page 137) the secants of any angles are always reciprocally in pro- 
portion to their cosines. 

2. The weight or load laid on each angle is directly proportional 
to the difference between the tangents of the elevations above the 
boriaon, of the two lines vrhich form the angle. 

3. The horiaontal thrust at every angle is the same constant quan- 
tity, and has the same proportion to Uie weight on the top of the 
q ppe raio st bar, as radius has to the taneent of the elevation of that 
W. Or, as the whole vertical a^ is to the line C H, so is the weight 
of the whole assemblage of bars, to the horizontal thrust. 

4. It may hence be deduced also, that the weight or pressure laid 
on any angle, is directly proportional to the continual product of the 
sine of that angle and of the secants of the elevations of the bare or 
lines which form it. 

Sekokum. This proposition is very fruitful in its practical conse- 
quences, and contains the whole theory of centerings, and indeed of 
arches, which may be deduced from the premises by supposing the 
constituting bare to become very short, like arch stones, so as to fonn 
the curve of an arch. It appeara too, that the horizontal thrust, 
which is constant or uniformly the same throughout, is a proper 
memsuring unit, by means of which to estimate the other thrusts and 
pressures, as they are all determinable from it and the given posi- 
tions ; and the value of it, as appeare above, may be easily computed 
from the uppermost or vertical part alone, or from the whole assem- 
blage together, or from t^y part of the whole, counted from the 
top downwards. 

In the most Important cases, a model of the structure may be 
made, and the relations of the pressures at any angle, whether hori- 
zontal, ^erticsl, or in Qie directions of the beams, may be determined 
by a spring steel-yard applied successively in the several directions. 



204 BQUILIBRIUM OP POLYGONS. [PABT II. 

2. If the whole figure in the preceding prohlem be inverted, or 
turned round the horizontal line AG (fig. 176) as an axis, till it be 
completely reversed, or in the same vertical plane below the first 
position, each angle D, dy &c., being in the same plumb line; and if 
weights t, k^ /, m, n, which are respectively equal to the weights laid 
on the angles B, C, D, E, F, of the first figure, be now suspended by 
threads from the corresponding angles h^ e^ d^ d^ f^ of the lower 
figure; those weights keep this figure in exact equilibrio, the same 
as the former, and all the tensions or forces in the latter case, 
whether vertical, horizontal, or oblique, will be exactly equal to 
the corresponding forces of weight, pressure, or thrust in the like 
directions of the first figure. 

This, again, is a proposition most fertile in its application, especi- 
ally to the practical mechanic, saving the labour of tedious calcula- 
tions, but making the results of experiment equally accurate. It 
may thus be applied to the practical determination of arches for 
bridges, with any proposed roadway ; and to that of the position of 
the rafters in a curb or mansard roof. 

3. Thus, suppose it were required to make such a roof, with a 
given width AE {^g, 177), and of four proposed rafters AB, BC, 
CD, DE. Here, take four pieces that are equal or in the same 
given proportions as those proposed, and connect them closely toge- 
ther at the joints A, B, C, D, E, by pins or strings, so as to be freely 
moveable about them; then suspend the whole from two pins, A, E, 
fixed in the same horizontal line, and the several pieces will arrange 
themselves in such a form, A^c^/E, that all its parts will come to 
rest in eauilibrio. Then, by inverting the figure, it will exhibit the 
form of the framing of a curb roof A B C D E, which will also be in 
equilibrio, the thrusts of the pieces now balancing each other, in the 
same manner as was done by the mutual tensions of the hanging 
festoon A6c</£. 

4. If the mansard be constituted of four equal rafters; then, if 
angle CAE = m, angle CAB=:;i?; it is demonstrable that 2 sin 
2 ;r = sin 2 m. So that if the span A E, and height M C, be given, 
it will be easy to compute the lengths AB, BC, &c. 

EwampU. Suppose A E = 24 feet, M C 12. 

IMF C* 

Then, —— = 1 = tan 45° angle C A M = m. 
MA 

.'. sin 2 ??» 1= sin 90° = 1, and sin 2 ;r = ^ 

.-. 2a; = 30°, and « = 15° =i CAB. 

Hence M A B = 45° -f 15° = 60° 

and M B A = J (180° — 2 x 15°) = 90° — 15° = 75° 

also A M B = 180° — (75° -h 60°) = 45° 

and lastly, sin 75° : sin 45 : : AM = 12 : AB = 8*7846 feet 

Note. — In this example, since AM = MC, as well as AB = BC, 

it is evident that AI B bisects the right anele A M C ; yet it seemed 

nreferable to trace the steps of a general solution. 



CHAP. II,] STABILITY OF ABCHBS. 205 

STABILITY OF ABCHBS. 

1 . If the effect of the force of gravity upon the ponderating matter 
of an arch and pier, he considered apart from the operation of the 
cements which unite the stones, &c., the investigation is difficult to 
practical men, and it furnishes results that require much skill and 
care in their application. But, in an arch whose component parts are 
united with a veir powerful cement, those parts do not give way in 
vertical columns, hut hy the separation of the ientire mass (including 
arches and piers), into three, or, at most, into four parts ; and in this 
case the conditions of equilibrium are easily expressed and applied. 

LetyF, /'F', (fig. 178) be the joints of rupture, or places at 
which the arch would most naturally separate, whether it yield in 
two pieces or in one. Let G be the centre of gravity of the semi- 
arch /F K *, and G' that of the pier A B F/. Let 'f I be drawn 
parallel to the horizon, and G H be demitted perpendicularly upon 
It ; also let G^ D be a perpendicular passing through G^, and F £ 
diBwn from F parallel to it. Then, 

2. The first case is when the arch fY Y' f tends to fall vertically 
in one piece, removing the sections J^F, f'Y'\ if W be the weight 
of the semi-arch fY K k^ and P that of the pier up to the joint y F, 
the equilibrium will be determined by these two equations : — viz., 

* • P = w 0? - *) (I.) 

."=-(n-^0 ("■' 

where ^ is the measure of the friction, or the tangent of the angle of 
repose of the material, and the first equation is that of the equili- 
Imnm of the horizontal thrusts, while the second indicates the equili- 
lirium of rotation about the exterior angle A of the pier. 

3. In the second case, when each of the two semi-arches Yk^k F\ 
tend to turn about the vertex k of the arch, removing the points 
F, F', the equilibrium of horizontal translation, and of rotation, will 
he respectively determined by the following equations : — viz., 

* . p = w (j^-i) (HI.) 

'H-C^-ID- (-, 

4. Hence it will be easy to examine the stability of any cemented 
arch, upon the hypothesis of these two propositions. Assume dif- 
ferent points, such as F in the arch, for which let the numerical 
values of the equations (I.) and (11.), or (III.) and (IV.) be com- 
puted. To ensure stability, the first members of the respective 
eaoations must exceed the second ; and those parts will be weakest 
where the excess is least 



A 

F£ 



206 



ARCHBS AND PIBB8. 



[part 



5. The following table extracted from Prof. Moseley's ^^ Mechanical 
Principles of Engineering," page 151, contains the value of the angle 
^ for the materials most usually employed in the construction of 
arches. 



Nature of Materials. 



GoeffidcBt 

of 

firiction. 



Lumiting 



Soft calcareous stone, well dressed, upon the same . 

Hard calcareous stone, ditto 

rommon brick, ditto 

Oak, endwise, ditto 

Wrought iron, ditto 

Hard calcareous stone, well dressed, upon hard cal- I 

careous stone I 

8oft, ditto 

Conamon brick^ ditto 

Oak, endwise, ditto 

Wrought iron, ditto 

Soft calcareous stone upon soft calcareous stone, with 1 

fresh mortar of fine sand ) 

EZPERTUENTS BT DIFFERENT OBSERVERS. 

Smooth freestone upon smooth freestone, dry. > 

(Rennie.) y 

Ditto, with fresh mortar. (Rennie.) 

Hard polished calcareous stone upon hard polished } 

calcareous stone • . . . . S 

Calcareous stone upon ditto, both surfaces being made \ 

rough with the chisel. (Bouchard!.) . . > 

Well dressed granite upon rough granite. (Rennie.) 
Ditto, with fresh mortar, ditto. (Rennie.) 
Box of wood upon pavement. (Ili^ier.) 
Ditto, upon {>eaten earth. (Herbert.) 
Libage stone upon a bed of dry clay 
Ditto, the clay being damp and soft 
Ditto, the clay being equally damp, but covered with 7 

thick sand. (Oreve.) ) 



74 
•76 
•67 
•63 
•49 

•70 

•75 
•67 
•64 
•42 

•74 



•71 



•68 

•78 

•66 
•49 
•68 
•33 
•61 
•34 

•40 



36 30 

36 62 

33 50 

32 13 



26 
35 



36 62 

33 60 

32 37 

22 47 

36 SO 



36 23 
33 26 
30 7 

37 68 

33 26 

26 7 
80 7 
18 16 

27 2 
18 47 

21 48 



If the section be drawn on smooth drawing pastehoard, upon a 
good sized scale, the places of the centres of gravity may he found 
experimentally, as well as the relative weights of the semi-arch and 
piers, and the measures of the several lines from the scale employed 
in the construction. 

If the dimensions of the arch were given, and the thickness of the 
pier required ; the same equations would serve ; and different thick- 
nesses of the pier might he assumed, until the first members of the 
equations come out largest. 

The same rules are applicable to domes, simply taking the un- 
gulas* instead of t^e profiles. 

* The ungulas^ as mentioned above, are the solids generated by the revolu- 
tion of the sections /FirK and /FA B about the vertical axis ArC. 



CflAP. II.] 8U8PENM0II BBIDOBB. SOT 

BQUILIBBIUH OF SUSPENSION BRIDGES. 

1. The several parts of a suspension bridge should always be pro- 
portioned in such a manner, that the tensile strain per square inch of 
section would be nnifbrm throughout the whole length of the chains ; 
and as the tension varies with the inclination of the chain, and to 
fulfil the above condition, the area of the chain must vary as the 
tension, it is q«ite evident that the chains of a suspension bridge 
ought not under any circumstances to be made of a uniform sectional 
area. 

If the chain was of a uniform section, and had only its own weight 
to carry, the form which it would assume would be a catenary; if on 
the other hand, the chain is supposed devoid of weight, and the load 
suspended fhnn it to be uniformly distributed horixontally, (as in the 
case of the roadway of a bridge,) then the form which the chain 
would assume would be the common parabola. Neither, however, of 
these supposed cases ever occur in practice, there being always three 
loads very differently distributed, viz., the weight of the chain itself, 
that of the roadway suspended from it, and that of the vertical rods 
by which the same is suspended ; consequently, the form which the 
chain when in equilibrium would assume is neither a catenary, nor 
a common parabola, but is between the two; and it approaches 
nearer to one or the other, as the weight of the chain or that of the 
roadvray predominates. Taking these three loads into accoun^ and 
varying the section of the chain so that its tensile strain is uniform 
tbroashout, the determination of the true form of the curve which it 
vodd assume when all its parts were in equilibrium, becomes a very 
eoiDplicated problem; it Ims, however, been very ably solved by 
Professor Moseley*, and the following formula which he has de- 
daoed, contains aU that is required for determinmg the form of the 
eorre end the dimensions of the chains. 

If fi^ = the weight of a bar of the material of the chain 1 square 
inch in section and 1 foot long, /a, :s the weight of a foot in length 
of the roadway, and, supposing the vertical suspending rods to be 
diflfbsed over the whole space between the chain and the road- 
]Bray, formins a uniform piate of such a thickness that its weight 
is precisely uie same as that of the actual suspending rods, let ^13 = 
the "weight of a square foot of this plate ; also, let K := the sectional 
%Tea in square inches of the chain at any point P (fig. 179), x and y 
being the ordinate and abscissa at the same point; let a = the 
Bemi-span, H = the deflexion, b = the length of the shortest sus- 
pending rod, € =3 the tension upon the lowest point of the chain, 

mMkd m c3 — ^^ where r represents the tenacity of the material of 

T 

Che chain per square inch, and m = the number of times that r 
exceeds the actual tensile strain upon the chains ; then 

* ** The Mechanical Principles of Engineering and Architecture,*' by the 
Her. Henry Af oaeley, page 647* 



208 SUSPENSION BBID0B8. [PART II. 

- = C.>-e'.Uj »-') <■•) 

"(iriir^y ("•) 

K _^^|2(,_ j)(!ii±!U + .) + ,}'.. (II,.) 

2. As, however, the use of these formulsB for determining the 
requisite numher of points in a hridge would be attended with much 
labour, it is only in cases where great accuracy is requisite that they 
need be employed ; for most purposes it will be sufficiently correct 
to assume that the form of the curve is a common parabola, in which 
case the formulse for determining the several elements of the bridge 
become much more simple, and easy of application. 

Let t ^ the coefficient of tension at the lowest point of the chain, 
or the quantity by which half the weight of the bridge must be 
multiplied to give the actual strain upon the chains, t^ = the coeffi- 
cient at the point of suspension, t^ := the mean coefficient of tension 
for the whole chain, « := the semi-span, d = the deflexion, x = 
the leoeth of half the chain, K^ = the sectional area of the chains 
at the lowest point, K^ = the same at the point of suspension, 
K3 := the mean sectional area of the whole chain, W := the weight 
of half the whole bridge, including the chain, y = the angle which 
the tangent to the curve at the point of suspension makes with the 
horizon, and b^ = the length of the suspending rod at any point P. 
The letters, /a^, /a^,, m, t, ^, :r, ^, and c, represent the same quan- 
tities as before. 

Then, the curve being a parabola, the length of the suspending rod 
at any point P may easily be found by means of the following 
property of the parabola ; 

y : rf : : a?' : «, 

therefore, supposing the roadway to be horizontal, we have 

h, =^ + V^ (IV.) 

And for the length of the chain, we have from the formula at 
page 129, for the length of parabolic arcs. 



=V 



•* + |rf' (V.) 



Also, from the method given at page 174, for drawing a tangent 
to any point in a parabola, we can easily deduce the value of 7, as 
follows: — if DE (fig. 180) be made equal to C D, the line drawn 
from A to £ will be the tangent fo the parabola at the point A, 
and therefore the angle CAE = 7, is determined from the proportion 
B : rad : : 2d : tan 7; 



CHAf. II.] SUSPENSION BRIDGES. 209 

.. tany = — (VI.) 

^iid sioce the chain A D B with the roadway suspended from it 
^ notbiog more than an equilibriated polygon, such as is described 
at page 202, having an infinite number of sides, all the relations 
there mentioned as existing between the several strains, may be 
applied to the present inquiry; therefore, if CE = 2rf be made to 
"^Pwsent the whole weight of half the bridge = W, then A C s= « 
^^^ represent the tension at D, and A E the tension at A ; therefore, 

''=2^ (V") 

^ = '-^ (VIII.) 

•nd <, = (I + «sc>) ^ (IX.) 

lien, for the mean sectional area of the chain, we have 
K,= ^^»' (X.) 

•"^d for the weight of the roadway and chain for half the bridge, 
W = K3X^j+fi2« (XL) 

*Tjen since /j W = c, the tension at the lowest part of the chain, 
^^ have for the sectional area of the chain at the same place, 

K, =^!lZ (XII.) 

T 

^^ K W being the tension at the point of suspension, the sectional 
**^ of the chain at that point will be 

K, =^^ ^Xjjj^ 

^ ^* In order to render the practical application of the foregoing 
^i^nle quite clear, we subjoin the following example. Let it be 
'^iriTed to determine all the elements of a suspension bridge, tlie 
•P*Ji of which is to be 360 feet, and the deflexion 30 feet; let the 
y^^} of the roadway for everj' foot in length = 4500 lbs., the 
^^**ciiy of a square inch of the chain = 67200 lbs., the weight of a 
^ 1 inch square and 1 foot long := 3*4 lbs., and the number of 
r^ that the cohesive strength of the iron is to exceed the 

"^ mbstituting these values for the letters representing the 
^«il qumiities, we obtain from formula (V.) 

p 



210 SUSPENSION BRIDGES. [Pi 

4 X SO'* 
180 + — = 183-33 feet, equal the length c 

the chain. 

From formula (VI.) 

2 X 30 1 

tan V = = - = -3333, which is the tangent* of li 

'^ 180 3 ' ^ 

the angle made hy the chains with the horizon at their pc 
suspension. 

From formula (VII.) 

180 

t, = = 3, the coefficient of tension at the lowest i 

* 2 X 30 
the chain; from formula (VIII.) 

180 X 10541 1 „,^^„ , ,, «• * r* 

/ = = 3*1623 equal the coefficient of teui 

2 2 X 30 ^ 

the point of suspension ; and from formula (IX.) 

180 
f, =(1 -f 1-0541) -. =30812 for the mean coeffici 

tension for the whole length of the chain. 

From formula (X.) 

4500 X 180 

67200 
— 1— — - - 183-33 X 3-4 
6 X 3-0812 

mean sectional area of the chain. 

From formula (XL) 

W = 269 X 183-33 x 3;4 + 4500 X 180 = 977,647 lbs. \ 
total weight of half the bridge; and finally, from formula (XI 
obtain 

6 X 3 X 977,674 ^^^ • u r *a. 

Kj = =262 square mches for the section 

of the chain in the centre ; and from formula (XIII.) 

ex 3-1623 X 977,674 ^^^ • t, r .u 

Kp = ^^^^^ = 276 square inches for the se 

* 67,200 ^ 

areA of the chain at the point of suspension. 

"• In order to find the angle of which '3333 ii the tangent, take tl 
'3333 a 7*522835, and adding 10 to the characteristic (for the reason < 
which see page 153), we have 9*522835, which we find from Table IV 
log tan of IS"" 26^, as above. 

t The secant of 18** 26' is obtained by a reverse process to that ezpl 
the foregoing note ; thus, from Table IV. we find the log sec of W\ 
10*022875, then subtracting 10 we have 0*022875, the number answ 
which in Table III. is 1 0541, the secant of 18** 26^ radius being 1. 



K3 = cpfa/\t\ ' — *^~ ^ ^^^ square inches f 



CHAP, in.] GBNBRAL DBFINITIONS. 211 



CHAP. III. 

DYNAMICS. 

Sect. I. General Dejtnitions, 

1. The man of a body is the q^uantity of matter of which it ift 
«wnp08ed; and is proportional to its weight, or to the^brc^ which 
nost be applied to the body to prevent its gravitating to the earth, 
i&d wbicb, being greater or less as the mass is greater or less, we 
i^u a measure of the mass itself. 

3. Demit^ is a word by which we indicate the comparative close- 

^ or otherwise of the particles of bodies, and is synonymous with 

^ term tpeeifie gravity. Those bodies which have the greatest num- 

^ of particles, or the greatest quantity of matter, in a given magni- 

^C) we call moA dense; those which have the least quantity of 

^*il(tt, leatt dejtse. Thus lead is more dense than freestone ; yre«- 

^^ more dense than oak ; and oak more dense than cork. 

. 3. The velocity with which a body in motion moves, is measured 

^y the space over which it passes in any given time; the unit usually 

^^Mned being one second. 

4. If the body passes over an equal space in each successive unit 
^ time, the body is said to move uniformly^ br to have a uniform 
^^ki^y and the measure of such velocity is the space actually passed 
^^er b? the body in each second. 

S* If, however, the body passes over a yreafer space in each sue- 

^^▼e second tlwn it did in the preceding, then it is said to move 

^tt ao aeceleraied vdocUy; when the differences between the spaces 

^^^ oter in any two successive seconds is the same, at whatever 

l^^nod of the body's motion they be taken, or in other words, when 

^ HKceaaive spaces form an arithmetical progression, the body is 

^^ to move with a uniformly accelerated velocity ; but when the 

^itts passed over in successive seconds increase according to any 

^^ law, the body is then said to have its velocity variably accele- 

^ If^ on the other hand, the body passes over a smaller space in 
^ SQoeeasive second than it did in the preceding, then it is said to 
^e with a retarded velocity; which, if the successive spaces form a 
^<c<eaaiig arithmetical series, is said to be uniformly retarded; if 
^^kerwiae, it is said to be variably retarded, 

p 2 



212 LAWS OF UNIFORM AND VARIABLE MOTION. [PART II. 

7. The velocity of a body whose motion is Tariable is expressed at 
any moment, by the space which it tcould pa^s ocer in a second^ if its 
Telocity at the moment spoken of were to continue uniform for that 
period. 

8. Mechanical effect is measured by the product of the nuus or 
weight of the body into the space over which it has been moved; no 
regard being had to the time occupied. The unit of mechanical effect 
employed in the subsequent pages, is a weight of 1 pound raised 
through a space of 1 foot, and is designated by the letter U. 

0. The momentum of a body in motion means the mechanical 
effect which such a body will produce in a moment (or second) of 
time, and varies as the weight of tlie body multiplied by its velocity, 

10. The vis viva of a body in motion is the whole mechanical 
effect which it will produce in being brought to a state of rest^ no regard 
being had to the time in which the effect is produced, and it varies as 
the weight of the body multiplied by the square of its vel4>city. 



Sect. II. On the general Latcs of Uniform and Variable Motion. 

1. As a proper understanding of the actual difference between the 
momentum and the vis viva of a body in motion, is esjtential to a 
correct application of the principles of dynamics, we shall take some 
pains to set this difference in as clear a light (for the student) as pos- 
sible. And it is of the more importance to do so, ns a diversity of 
opinion upon this subject has existed amongst some of the most emi- 
nent mathematicians, and much time and talent has been expended 
by them in supporting errors w Inch have arisen entirely in a miscon- 
ception of terms, ai)d in excluding from their conclusion the con- 
sideration of time which they included in their premises. 

2. From carefully conducted and often repeated experiments, the 
following results with regard to bodies in motion have been ob- 
tained : — 

I. If a body of a certain weight, and moving with a given 
velocity, meet another body of double that weight, and 
moving with half the velocity, the two bodies will destroy 
each other's motion, and both will be brought to a state of 
rest. 

II. A body of a certain weight and moving with a given velo- 
city, being subject to a uniformly retarding forc^, (i . e, a uni- 
form force acting constantly in a contrary direction to the 
body's motion,) will move over a certain space in being 
brought to rest, and will occupy a certain time in doing so ; 
then another body of the same weight, but moving with half 
the velocity of the former, being Subject to the same uni- 
formly retarding force, will move over one quarter of the 



CHAP. III.] LAWS OF UNIFORM AND VARIABLB MOTION. 213 

space moTed over by the former, in being brought to a state 
of rest, and will occupy in doing so ^o^tbe time. And an- 
other body of the same weight, but moving with one-third of 
the velocity of the first, will move over one-ninth of the space, 
and occupy one-third the time of the first, in being brought to 
a state of rest. 

Now the diversity of opinion to which we have alluded above, has 
arisen from the (at first sight) apparently contradictory nature of 
these two results : one party has drawn a conclusion from the first 
experiment that the force of a body in ynotion is directly <w its velo- 
city; and the other party has drawn a conclusion from the second 
experiment that \he force of a body in motion is directly as the square 
of its velocity. These errors have arisep from the term " force of a 
body in motion" being used without any fixed and definite meaning 
being attached to the same. 

3. The proper measure of the whole force (i. e. vis viva) of a body 
in motion, is the mechanical effect, or (as Professor Moseley very 
significantly terms it) the work^ which it is capable of performing in 
being brought to a state of rest. That evidently being the force due 
to its faction which is required to destroy the same^ and which is 
directly as the square of its velocity^ as found by the second experi- 
ment. 

4. If, however, we only consider the mechanical effect (or the 
work) which a body in motion is capable of performing in a given 
time, (i.e. its momentum^) we shall find from the second experiment, 
that ajthough the body having twice the velocity ultimately produced 
four times Qie effect, in doing so it occupied just twice the tii>c; and 
^in, although the body moving with three times the velocity tUti- 
niatefy produced nine times the effect, it occupied in doing so three 
times the time ; and, therefore, the mechanical effect produced in a 
yicen time by the bodies was directly as tlie velocities. And this con- 
clusion, drawn from the second experiment, is in accordance with 
that which must be drawn from the first experiment, in which it is 
eyident that the effect produced by both bodies must be in the same 
(although indefinitely small) time. 

5. The force of gravity being constantly the same, both in amount 
%iid direction, and being practically uniform* in its action for such 
^niall distances from the earth's surface as come under consideration 
iQ ordinary dynamical investigations, has been universally adopted as 
the onit of measure for all other forces. 

6. The actual amount of the force of gravity is measured by the 
Velocity which a body falling in vacuo, in the latitude of London, will 

* The force of gravity varies as the square of the distance from the earth*s 
oencre, and therefore becomes less as we asueiid alxive the surface of the earth, 
in the proportion of I to *9994, a difference too small to require notice in any 
question of terrestrial mechanics. 



214 MOTION UNIFORMLY ACCBLBBATED. [PABT II. 

acqaiie in one second of time, and which, hy carefully conducted ex- 
peiiments, has heen ascertained to he 386*289 inches, or about 32^ 
feet per second^ and this latter quantity is usually represented by the 
symbol g. 

MOTION UNIFORMLY ACCBLERATED. 

1 . If we now put W to denote the quantity of matter, or weigLt 
of a body in motion, v its velocity in feet per second, M its momen- 
tum, and V its vU viva^ both expressed in units of mechanical effect, 
or pounds raised through a space of 1 foot; we have 

M = Wfj (I.) 

^ = ^2-^ <"•> 

2. And further, if s h? put to represent the space passed over in 
the time ^, by a body subject to the uniform forced*; then we have 
the following relations between all these quantities : — 

V = «W = -W=-yW = -2/W . (III.) 

2 22/ ^^'> 

«=//=—= ^2fB .... (VI.) 

' = ) = v-a/7 •• <™> 



MOTION OF BODIES UNDER THE ACTION OF GRAVITY. 

1. When the uniform force is that of gravity,/ := g; and g being 
proportional to the weight of the body (W), the foregoing relations 
are simplified, and then become as follows : — 

, = ^ = ^J! = ^ (IX.) 

2 2 2^ ^ ^ 

V^gt^— ^^/Yf% (X.) 



CHAP. III.] DE8CBNTS BY OBAVITV. 215 



= • = — = /u 



(XI.) 



-=r-7' = n '-■) 

2. Any two of these quantities being given, the other two may be 
immediately ascertained from the above equations; the following 
table shows their actual numerical values for the first four seconds of 
the motion of a heavy body filling freely by the action of gravity : — 

The times in seconds being 1", 2", 3", 4", &c. 
The velocities in feet will be 32^, 64 j, 96i, 128^, &c. 
The spaces in the whole times l6-j^^ 64|, 144}, 257|, &c. 
And the space for each second 16^, 48^, 80j^, 112^*^, &c. 

of which spaces the common difference is 32^ feet, equal y, the mea- 
sure of the force of gravity. 

3. If, instead of a heavy bodv being allowed to fall freely, it be 
propelled vertically upwards or downwards with a given velocity, v, 
then 

M^tvzfi^; (XIII.) 

aa ezpressioD in which the upper sign — must be taken when the 
projection is upwards^ the lower sign + when the projection is down- 
wardM. 

4. When only an approximate result is required with reference to 
bodies falling vertically, 32 may be put for ^, instead of 32 J : there 
would then result, in motions from quiescence, 

'-'"■-a"? • ■ • ■ (»"'•) 

t;=8v^i=: — = 32^ (XVI.) 

^lius, if the space descended were 64 feet, we should have i; := 8 

Q 

^ B = 64 feet per second, and ^ = - = 3 seconds. 

If the space descended were 400; then t? = 8 x 20 = 160 feet 

l-^,e«o.d,«.d. = ¥ = 6eeco»dB. 
4 

5- The force of gravity differs a little at different latitiules ; the law 
^ the variation is not as yet precUely ascertained ; but the following 



216 MOTION OVEB PULLEYS, ETC. [PART II. 

theorems are known to represent it very nearly. That is, if ^ denote 
the force of grayity at latitude 46% ^j the force at the poles, ^^ the 
force at the equator, and ^3 the force at any other place : then 

^, =^(1 + -002837) (XVII.) 

g^z=g{\ — 00-2837) (XVIII.) 

g^^g (1- — -002837 cos 2 lat.) . . (XIX.) 



MOTION OVER A FIXED PULLEY. 

In this case let the two weights which are connected hy the cord 
that goes over the pulley be denoted by W . and W , : then — ^ — — ^ 9 
= W/in formula (III.); so that 

W,-W, gj_ 
W, +W, ' 2 

Or, if the resistance caused by the rigidity of the rope, and the fric- 
tion and inertia of the pulley, be represented by r; then 

'-W,+W, + r • 2 ^^^-^ 

Example 1. Suppose the two weights to be 6 and 3 lbs. te&Y*^^' 
ively, what will be the space descended in 4 seconds ? 

16 = 16jV X 4 = 64jfeet. 

Example 2. But suppose that, in an actual experiment witl» ^ 
weights of 6 and 3 lbs. over a pulley, the heavier weight desce-^^ ^^ 
only 50 feet in 4 seconds. 

the same in both examples, 

we have w, + W^ + r : W. + W, :: 64J : 60 
or, dividendo r : W, -f W^ :: 14J *: 60 
that is, r : 5 -f 3 :: 14J : 60 

whence r = ^^tJ^ = ^4^^ = 22933 lbs. 
50 50 

the measure of the resistance and the inertia. 

2. Similar principles are applicable in a variety of other caies / 
and by varying the relations of W„ W^, and r, the force may hvre 



CHAP. III.] MOTION ON INCLINED PLANES. 21? 

any assigned ratio to that of gravity; which is, indeed, the founda- 
tion of Mr. Atwood's elegant apparatus for experiments on accele- 
rating forces; an account of which may he seen in the 2nd volume of 
my Mechanics^ or in almost any of the general dictionaries of arts 
and sciences. 

3. If, instead of pulleys, small wheels and axles, as in figure 181, 
be employed^ to raise weights by the preponderance of equal weights: 
then, if the diameter of the wheel and axle A be as 3 to 2 ; those of 
the wheel and axle B, as 5 to 2; and those of C, as 8 to 2; it will 
be found that the weight b will be elevated more rapidly than either a 
or c : the proportion of 5 to 2, (or, more accurately, of 1 -f- ^^2 to I) 
being in that respect the most favourable. 

MOTION ON INCLINED PLANES. 

1 . When bodies move down inclined planes, the accelerating force 
(independently of the modification occasioned by the position of the 
centre of gyration) is equal to g multiplied by the quotient of the 
beight of the plane divided by its length, or what is equivalent, by 
the sine of the inclination of the plane. In this case, therefore, put- 
ting « for the inclination of the plane, the formulae become 



(XXI.) 



_ ^ ^ . sin I ^ tv 

2 2^ . sin f "" 2 

2 9 
r = y / . sin I = >/ {^g9 . sin t) = — (XXII.) 

*=./-^'-=^' (xxm.) 

Purther, if », be the velocity with which a body is projected up or 
"Oivn a plane, then 

c = Vj ip y / . sin I (XXIV.) 

^^ . sin f v^ — v^ , 

s=^v^t^^ =-~^ —. . . (XXV.) 

* ^ 2 2^ . siuf ^ ^ 

Alaking v, = 0. in equation (XXIV.), and the latter' member of 
^^ Nation (XXV.), the first vnll give the lime at which the body will 
'^^'^^e to rise, the latter the spctce. 

Example. Suppose a body be projected up a smooth inclined 
p^Be whose height is 12 and length 193 feet, with a velocity of 20 
.^^t per second, how high will it rise up the plane before its motion 
^^ extinguished? 

Then substituting these values in formula (^XXV.), wc have 

400-0 400 400 ,_^ , . , 

"= 64J X ^»/, =193 12 =-^ = 100fcet,thcspacerequired. 

3 ^ 193 



218 MOTION ON INCLINED PLANES. [PART II. 

2. With regard to the velocities acquired by bodies in falling down 
planes of the same height, this proposition holds ; viz. that they are 
all eaual, estimated in their respective directions. Thus, if AD, BE, 
CF, (fig. 182), be planes of different inclinations, and AC, DF, hori* 
zontal lines, the balls A, B, C, after descending along those planes 
will have equal velocities when they arrive at the points D, £, F, re- 
spectively. 

3. Also, all the chords, such as AD, BD, CD, (fig. 183), that ter- 
minate either in the upper or the lower extremity of the vertical dia- 
meter of a circle, will be described in the same time by heavy bodies 
A, B, C, running down them; and that time will be equal to die time 
of vertical descent through the diameter DE. 

4. If three weights, asA, B, C, (fig. 184), be drawn up three 
planes of different inclinations, by three equal weights hanging from 
cords over pulleys at P, then if the length of the middle plane be 
ti0ice its height^ tLe body B will be drawn up that plane, quicker than 
either of the other weights A or C. Or, generally, to ensure an ascent 
up a plane in the least time, the length of the plane must be to its 
heighty as ttoice the weight to the power employed. 

5. If it be proposed to construct a roof over a building of a given 
width, so that the rain shall run quickest off it, then each side of the 
roof must be inclined 45° to the horizon, or the angle at the ridge 
must be a right angle. 

6. The force by which spheres, cylinders, &c. are caused to re- 
volve as they move down an inclined plane (instead of sliding) is the 
adhesion of their surfaces occasioned by the pressure against the 
plane : this pressure is part of the body's weight ; for the weight 
being resolved into its components, one in the direction of the plane, 
and the other perpendicular to it, the latter is the force of the pres- 
sure upon the plane; and, while the same body rolls down the plane, 
will be expressed by the cosine of the plane's elevation. Hence, 
since the cosine decreases while the arc or angle increases, after the 
angle of elevation ariives at a certain magnitude, the adhesion may 
become less than what is necessary to make the circumference of the 
body revolve fast enough ; in this case the body descends partly by 
sliding and partly by rolling. And the same may happen in smaller 
elevations, if the body and plane are very smooth. But at all eleva- 
tions the body may be made to roll by Uie uncoiling of a thread or 
riband wound about it. 

If W denote the weight of a body, a the space described by a body 
falling freely by the action of gravity, or sliding freely down an in- 
clined plane, then the spaces («,) described by rotation in the same 
time by the following bodies, will be in these proportions. 

(1.) In a hollow cylinder, or cylindrical surface, «| = ^ « and the 
tension of the cord, in the first case = | W. 

(2.) In a solid cylinder, «, = ^ «, and the tension = ^ W. 

(3.) In a spheric surface, or thin spherical shell, «, = |^ «, and the 
tension = | W. 

(4.) In a solid sphere, «| =: -^ «, and the tension = -^ W. 




CHAP, in.] MOTION ON INCLINBD PLANES. 210 

If two cylinders be taken of equal size and weight, and with equal 
protuberances upon which to roll, as in the mar- 
ginal figures: then, if lead be coiled uniformly 
over the curve surface of B, and an equal quan- 
tity of lead be placed uniformly from one* end 
to the other near the axis in the cylinder A, that 
cylinder will roll down any inclined plane quicker 
than the other cylinder B. The reason is that 
each partide of matter in a roUing body reiisU 
motion in proportion to the squarb of its distance 
JrofH ike axis of motion ; and the particles of lead 
which most resist motion are placed at a greater distance from the 
axis in the cylinder B than in A. 

7. The friction between the surface of any body and a plane, may 
be eanly ascertained by gradually elevating the plane untiJ the body 
upon it/iM/ begins to dide. The friction of the body is to its weight 
as the height of the plane is to its base, or as the tangent of the in- 
clination of the plane is to the radius. Thus, if a piece of stone in 
-weight 8 pounds, just begins to slide when the height of the plane is 
2 feet, and its baise 2\ ; then the friction will be ^ the weight, or 4 
of 8 lbs. = 6| lbs. 

8. After motion has commenced upon an inclined plane, the fric- 
tion is osoally much diminished. It may easily be ascertained expe- 
Timentaiiy, by comparing the dme occupied by a body in sliding down 
a plane of given height and length, or given inclmation, with that 
which the simple theorem for /, (XXIII.) would give. For, iff he 
the value of the friction in terms of the pressure, the theorem for 
the time wiU be 

^1 = A / — T-' "^y instead off = a / : — • Hence 

r,«:<«::sini:sini-/ . . . (XXVI.) 

Example. Suppose that a body slides down a plane in length 30 
feet, height 10, in 2^ seconds, what is the amount of the friction? 

Here f « A / ^ = A /;;7n r = 2-366 nearly. 

Hence (2-6)» : (2-366)' :: § : -27603 = sin • -/ 

Consequently, -33333 — 27603 = -0573 value of the friction, the 
weight being unity. 

9. When a weight is to be moved either up an inclined plane, or 
along on horizontal plane, the angle of traction PWB (fig. 185) that 
the weight may be drawn with least effort, will vary with the value 
off. The magnitude of that angle PWB for several values of f are 
exhibited in the following table: — 



220 



MOTIONS ABOUT A PIXBD CBNTBR OB AXIS. [PABT IT. 



/ 


PWB 


1/ 


PWB 


/ 


PWB 


/ 


PWB 


/ 


PWB ij/ 


PWB 


1 


45° 0' 


II 


26°34' 


^ 


18^26' 


i 


14° 2' 


t 


11°19! ^ 


9^28' 


t 


38 40 


23 58 


^ 


16 54 


tV 


13 15 


i\ 


10 47 IJ4 


8 8 


^ 


33 41 


^ 


21 48 


-f 


15 57 


^ 


12 32 


A 


10 18 


4 


7 8 


+ 


29 45 


A 


19 59 


-h 


14 56 


A 


11 63 


A 


9 52 


i 


6 20 



10. If, instead of seeking the line of traction so that the moving 
force should he a nainimuro, we required the position such that the 
suspending force to keep a load from descending should he a mini- 
mum, or a given force should oppose motion with the greatest energy; 
then the angles in the preceding tahle will he still applicahlc, only 
the angle in any assigned case must he taken helow, as BWp. This 
will serve in the huilding and securing walls, in the construction of 
hanks of earth, fortifications, &c., and in arranging the position of 
land-ties^ &c. 



Sect. III. Motions abotU a fixed Center or Axis. 
Centers of Oscillation and Percussion. 

1. The center ofosciUation is that point in the axis of suspension 
of a vihrating hody in which, if all the matter of the system were 
collected, any force applied there would generate the same angular 
velocity in a given time as the same force at the center of gravity, the 
parts of the system revolving in their respective places. 

Or, since the force of gravity upon the whole hody may be con- 
sidered as a single force (equivalent to the weight of the hody) 
applied at its center of gravity, the center of oscillation is that point 
in a vihrating hody in which, if the whole were concentrated and 
attached to the same axis of motion, it would then vihrate in the 
same time that the hody does in its natural state. 

2. From the first definition it follows that the center of oscillation 
is situated in a right line passing through the center of gravity, and 
perpendicular to the axis of motion. It is always farther from the 
point of suspension than the center of gravity. 

3. The center of percussion is that point in a hody revolving about 
an axis, at which, if it struck an immovable obstacle, the whole of 
its motion would be destroyed, or it would not incline either way. 

4. When an oscillating body vibrates with a given angular velo- 
city, and strikes an obstacle, the effect of the impact will be the 
greatest if it be made at the center of percussion. For, in this case 
the obstacle receives the whole revolving motion of the body; 
whereas, if the blow be struck in any other point, a part of the 
motion of the body will he employed in endeavouring to continue the 
rotation. 

5. If a body revolving on an axis strike an immovable obstacle 



CHAP. III.] PENDULUMS. 221 

at the center of percussion, the point of suspension will not he 
affected hy the stroke. We can ascertain this property of the point 
of suspension when we give a smart hlow with a stick. If we give 
it a motion round the joint of the wrist only, and, holding it at one 
extremity, strike smartly with a point considerahly nearer or more 
remote than ^ of its length, we feel a painful wrench in the hand : 
but if we strike with that point which is precisely at ^ of the length 
(that being the situation of the center of percussion), no such dis- 
agreeable strain will be felt. If we strike the blow with one end of 
the stick, we must make its center of motion at ^ of its length from 
the other end; and then the wrench will be avoided. 

6. The distance of the center of percussion from the axis of motion 
is equal to the distance of the center of oscillation from the same : 
supposing that the center of percussion is required in a plane passing 
through the axis of motion and the center of gravity. 

SIMPLE AND COMPOUND PENDULUMS. 

1 . A simple pendulum^ theoretically considered, is a single weight, 
regarded as a point, or as a very small globe hanging at the lower 
extremity of an inflexible right line, void of weight, and suspended 
from a fixed point or center, about which it oscillates. 

2. A compound pendulum is one that consists of several weights 
movable about one common center of motion, but so connected 
together as to retain the same distance both from one another and 
from the center about which they vibrate. 

Or any body, as a cone, a cylinder, or of any shape, whether 
regular or irregular, so suspended as to be capable of vibrating, may 
be regarded as a compound pendulum ; and the distance of its center 
of oscillation from any assumed point of suspension, is considered as 
the length of an equivalent simple pendulum. 

3. If O represent the distance of the center of oscillation from the 
point of suspension, and G the distance of the center of gravity from 
the same point, it has been found that 6 O is a constant quantity, for 
the same body and the same plane of vibration. 

4. Any such vibrating body will have as many cefiters of oscillation 
as yon give it points of suspension: but when any one of those 
centers of oscillation is determined, either by theory or experiment, 
the rest are easily found by means of the foregoing property that O G 
is a constant product, or of the same value for the same body. 

5. When a body either revolves about an axis, or oscillates, the 
sum of the products of each of the material elements, or particles of 
that body, into the squares of their respective distances from the axis 
of rotation, is called the momentum of inertia of that body. 

6. A point, or very small body, on descending along the successive 
sides of a polygon in a vertical plane, loses at each angle a part of 
its actual velocity equal to the product of that velocity into the versed 
sine of the angle made by the side which it has just quitted, and the 



222 PENDULUMS. [part II. 

prolongation of the side upon which it is just entering. Therefore, 
that loss is indefinitely small in curves, 

7. A heavy body which descends by the force of gravity along a 
curve situated in a vertical plane, has, in anv point whatever, the 
same velocity as it would have had if it had fallen through a vertical 
line equal to that between the top and the bottom of the arc run 
over : and when it has arrived at the bottom of any such curve, if 
there be another branch either similar or dissimilar, rising on the 
opposite side, the body will rise along that branch (apart from the 
consideration of friction) until it has reached the horizontal plane 
from which it first set out. Thus, after having descended from A to 
V (fig. 186), it will have the same velocity as that acquired by 
falling through D V, and it will ascend up the opposite branch until 
it arrives at B. 

8. If the body describe a curve by a pendulous motion, the same 
property will be found to obtain, setting aside the effects of friction. 
Thus, let a ball hang by a flexible cord SD (fig. 187) from a pin 
at S : then, after it has descended through the arc DE, it will pass 
through an equal and similar arc £A, going up to A in the same 
horizontal line with D, and ascending from £ to A in an interval of 
time equal to that which it descended from D to £. But, if a pin 
projecting from P or p stop the cord in its course, the ball will still 
rise to B or to C, in the same horizontal line with A and D ; but will 
describe the ascending portions of the curve in shorter intervals of 
time than the descending branch. 

9. When a pendulum is drawn from its vertical position, it will be 
accelerated in the direction of the tangent of the curve it would 
describe, by a force which is as the sine of its angular distance from 
the vertical position. Thus, the accelerating force at A (fig. 188), 
would be to the accelerating force at B, as A F to BE. (See art. 5, 
on the Center of Gravity,) This admits of an easy experimental 
proof. 

10. If the same pendulous body descend through different arcs, iu 
velocity at the lowest point will be proportional to the chords of the 
whole arcs described. Thus, the velocity at D, after passing through 
ABD, will be to the velocity at D after descending through ^e 
portion BD only, as AD to BD. 

1 1 . Farther, the velocity after describing ABD, is to the velocity 

after describing BD, as v^FD is to v^ED. If, therefore, we would 

impart to a body a given velocity V, we have only to compute the 

V* V* 
height FD, such that FD = — = — r- feet, and through the point 

F draw the horizontal line FA; then, letting the body descend as a 
pendulum through the arc ABD, when it arrives at D it will have 
acquired the required velocity. This property is extremely useful in 
experiments on the coUision of bodies. 



CHAP. III.] PENDULUMS. 223 

12. The oscillations of penclulums in any arcs of a cycloid are 
ifockrofud^ or performed in equal times. 

13. Oscillations in »n)aU portions of a circular arc are nearly 
iiockronal, 

14. The numbers of oscillations of two different pendulums, in 
the same time, and at the same place, are in the inverse ratio of the 
square roots of the length of those pendulums. 

15. If / be the length of a single pendulum, or the distance from 
the point of suspension to the center of oscillation in a compound 
pendulum, g = the measure of the force of gravity (32^ feet, or 
386 inches at the level of St. Paul's* in the latitude of London), 
/ the time of one oscillation in an indefinitely small circular arc, 
And w =s 3*141593 : then 



=-v-: <■■' 



16. Conformably with this we have 

39 \ inches, length of the second \ pendulum 

9Jf inches half second in the 

4^ inches third of second latitude 

2-^^ inches quarter second) of London. 

17. Putting y, for the force of gravity in any latitude and at any 
^titude, we have also /= -20264 x |^| 

and 1^1 = 4-9348/ (IL) 

In other words, whatever be the force of gravity, the length of a 
second pendnlum, and the space descended freely by a falling body 
in 1 second, are in a constant ratio, 

18. If /, be the length of a pendulum, ^, the force of gravity, 
and /, the time of oscillation at any other place, then 



'• ■■■■ ^/i ■■ Vf. '■" 



) 



If the force of gravity be the same, 

t :t,:: ^l: ^ I, (IV.) 

If the aune pendulum be actuated by different gravitating forces, 
we have 

* At the level of ike mo, in the latitude of liondon, g is 386-289 inches, and 
the c o rre ^ w m dipg length of the necond penduhim is 39*1393 inches, according 
to the determination of Major Kater. Conformably with this result are the 
Biunbers in the Table (in the Appendix), computed at the expense of Messrs. 
BrawnA and DoniUn, and obli^nngly communicated by them for this work. 
It has been suspected by M. Beuel'^ and demonstrated by Mr. Francis BaUy^ 
that, in the refined computations relative to the pendulum, the formulae for the 
redaction to a vacuom are inaccurate, and that, in consequence, we do not yet 
pneit^ know the length of a second pendulum. See Phil. Transac. 1832. 



224 PENDULUMS. [part II. 

When pendulums oscillate in equal times in different places, 
we have 

gig, :: I : l^. 

For the variations of gravity in different latitudes, see formulie 
(XVII., XVIII., XIX.) in the preceding section. 

18. If the arcs are not indefinitely short, let v denote the versed 
sine of the semi-arc of vibration ; then 

t = , y^i(l + i r + ^j 0- + &c) .... (VI.) 

In which, when the serai-arc of vibration does not exceed 4 or 5 
degrees, the third term of the series may be omitted. 

If the time of an oscillation in an indefinitely small arc be 1 second, 
the augmentation of the time will be 

for a serai-arc of 30° 001675 

of 15^* 000426 

of 10° 0-00190 

of 5° 0-00012 

of 2J° 000003 

So that for oscillations of 2^° on each side of the vertical, the 
augmentation would not occasion more than 2^' difference in a day. 

19. If D denote the degrees in the semi-arc of au oscillating 
pendulum, the time lost in each second by vibrating in a circle 

instead of the cycloid, is ; and consequently the time lost in 

a whole day of 24 hours, or 24 x 60 X 60 seconds, is ^ D^ nearly. 
In like manner, the seconds lost per day by vibrating in the arc of 
A degrees, is J A^. Therefore, if the pendulum keep true time in 
one of these arcs, the seconds lost or gained per day, by vibrating in 
the other, will be 4 (I>~ — A^). So, for example, if a pendulum 
measure true time m an arc of 3 degrees, it will lose 11 1 seconds 
a day by vibrating 4 degrees ; and 26| seconds a day by vibrating 
5 degrees : and so on. 

20. If a clock keep true time very nearly, the variation in the 
length of the pendulum nccessaiy to correct the error will be equal 
to twice the product of the length of the pendulum, and the error 
in time divided by the time of observation in which that error is 
accumulated. 

If the pendulum be one that should beat seconds, and ^, the daily 
variation be given in minutes, and n be the number of threads in an 
inch of the screw which raises and depresses the bob of the pendulum, 

then X = =■ — ^- i = '05434 n ^ = X « 'i » nearly, for the 

24 X 60 ^^ ^ 

number of threads which the bob must be raised or lowered, to make 

the pendulum vibrate truly. 

21. For civil and military engineers, and other practical men, it is 



CHAP. III.] CENTEB OF OSCILLATION. 225 

highly useful to have Vk portable pendidum^ made of painted tape with 
a brass bob at the end, so that the whole, except the bob, may be 
rolled up within a box, which may be enclosed in a shagreen case. 
The tape is marked 200, 190, 180, 170, 160, &c., 80, 75, 70, Qli, 
OO, at points, which being assumed respectively as points of suspen- 
Qon, the pendulum will make 200, 190, &c., down to 60 vibrations 
in a minute. Such a portable pendulum may be readily employed 
in experiments relative to falling bodies, the velocity of sound, &c. 

22. If the momentum of inertia (§ .5, page 221) of a pendu- 
lum, whether simple or compound, be divided by the product of 
the pendulum's weight or mass into the distance of its center of 
gravity from the point of suspension for axis of motion), the quotient 
will express the distance of the center of oscillation from the same 
point (or axis). 

23. Whatever the number of separate masses or bodies which 
constitute a pendulum, it may be considered as a single pendulum, 
Dvhose center of gravity is at the distance d from the axis of suspen- 
sion, or of rotation : then, if K' denote the momentum of inertia of 
tfmt body divided by its mass, the distance O from the axis of 
rotation to the center of oscillation, or the length of an equivalent 
mraple pendolnm, will be 

= £±^.. (VII.) 

a 

24. To find the distanoe of the center of oscillation from the point 
^^w axis of suspension, experimentally. Count the number, n, of 
^^ecillations of the body in a very short arc in a minute ; then 

O^l^ (VIII.) 

Tims, if a body so oscillating made 50 vibrations in a minute; 

^'^O^-j^^SS'S^ inches. 

Or, O = 89 j^ fy in inches, / being the time of one oscillation in a 
^ry small arc. 

If tlie arc be of finite appreciable magnitude, the time of oscilki- 
^n must be reduced in the ratio of 8 + versm of semi-arc to 8, 
'ore the rule is applied. 

25. From the foregoing principles are derived the following ex- 
sessions for the distances of the centers of oscillation for the several 

suspended by their vertices and vibrating flatwise, vis. :-— 
(1.^ Bi^t line or very thin cvlinder, O = | of its length. 
(2.) IsMceles triangle, O 3= | of its altitude. 
(3.) Circle, O == | radius. 
(4.) Common parabola, = 4^^^ altitude. 

* For tome curiout and vahiabla theorems, by Professor ^try, for the re- 
^taecioo oi viknitSani ia the air to those in a VAoiuiin, tee Mr. F. Bailya peper 
*>rerTed to in the preoeding note. 



22i} CKNTKR OF USCILLATION. [PAKT II. 

(5.) Any parabola, O = X its altitude. 

3 m + 1 

Bodies vibrating laterally or sideways, or in their own plane : 

(6.) In a circle, O = } of diameter. 

(7.) In a rectangle suspended by one angle, O = | of diagonal. 

(8.) Parabola suspended by its vertex, O = 4 a^tis + J parameter. 

(9.) Parabola suspended by middle of its base, O = ^ ^^^^ ~^ i 
parameter. 

,,^ N T r . 1 3arc X rad 

(10.) In a sector of a circle, O = = — = — . 

^ 4 chord 

/,, X T ^ - • (ra<l of base)* 

(11.) In a cone, O == 4 axis 4- -^ ; — . 

^ -^ 5 axis 

2 rad^ 

(12.) In a sphere, O = rad -f rf -f ---= j: ; where d is the 

5 (a + rad) 

length of the thread by which it is suspended. 

(13.) If the weight of the thread is to be taken into the account, 
we have the following distance between the center of the ball and 
that of oscillation, where B is the weight of the ball, d the distance 
between the point of suspension and its center, r the radius of the 
ball, w the weight of the thread or ^ire, and 1 the distance of the 
center of oscillation from the center of gravity, we have 

^^ aw^^B)4r''^}w(2dr±d^ 

(|«j-fB)</ — r» ^ "^ 

Or, if B be expressed in terms of w considered as a unit, then 

^ = ^^ (X.) 

(14.) If two weights W, W,, be fixed at the two extremities of a 
rod of given length, S being the center of motion between W and 
W, ; then, if d equal the distance of the weight W from S, D equal J 
the distance of W, from S, and m the weight of a unit in length of ^ 
the rod, we shall have 

^ ■" wD« + 2W,D — m</- — 2W</ ^ *^ 

the radii of the balls being supposed very small in comparison witbrJ 
the length of the rod. 

(15.) In the bob of a clock pendulum, supposing it two equaT^ 
spheric segments joined at their bases, if the radii of those bases b^^ 
each = ^, the height of each segment c, and d the distance from th^ j 
point of suspension to the center of the bob, then is 

'=w- ''^'AV/" <^"-) 

which shows the distance of the center of oscillation below the ccnteas'J 
of the bob. 



CHAP. III.] CENTER OF OSCILLATION. 227 

If r the radius of the sphere he known, the latter expression 
becomes 

'- "t:^^" <™'-' 

(16.) Let the length of a rectangle be denoted by /, its breadth by 
2 Wy the distance (along the middle of the rectangle) from one end to 
the point of suspension by ^ then the distance O, from the point of 
mspensioD to the center of oscillation, ^i-ill be 

whether the 6gare be a mere geometrical rectangle, or a prismatic 
metallic plate of uniform density. It follows from this theorem, that 
a plate of 1 foot long and | of a foot broad, suspended at a fourth of 
% foot from either end, would vibrate as a half second pendulum. 
AIMS tliat a plate a foot long, -^^ of a foot wide, and suspended at 
1^ of m foot from the middle, would vibrate 30,469 times in 5 hours. 
Jknd henoBy ike length of a foot may he determined experimentcUly 
hif vibratume. 

(17.) If a thin rod» say of a foot in length, have 
& ball of an inch diameter at each end, A and B, 
&nd a moveable point of suspension, S; then the 
tame of oteiUation of such a pendulum may be 
wnade as long as we please ^ by bringing the point 
of suspension nearer to the middle of the rod. 

Or, if tbe point of suspension be fixed, the dis- 
tance O (and consequently the time of oscillations 
'^'phicb is as >/0) may be varied by placing A 
vi«arer to or farther from 8. And this is the prin- 
ciple of the Metronome^ by wliich musicians some- 
times regulate their time. 



(18.) If the weight of the connecting rod be evanescent witli 
^e^ard to the weight of the balls A and B ; then if R equal the radius 
^f the larger ball, r that of the smaller, D and d the distances of their 
■■'oipcctivc centers from 8 : we shall have 

^__ B«(5D» + 2R^)-hrM5(/''»H-2r^) 

^ 5(DR»)-.rfr') ^^^'^ 

When R and r are equal, this becomes 

O = (D + rf) + ^ . ^^ (XVI.) 

(19.) If the minor and major axes of an ellipse (or of an elliptical 
P«e of wood or metal) be as 1 to >/ 3, or as 1000 to 1732 ; then, 

Q 2 




228 COMPENSATION TRNDULUMS. [PART IF. 

if it be suspended at one extremity of the minor axis, the center of 
oscillation will be at the other extremity of that axis, or its oscilla- 
tions will be performed in the same time as those of a simple pendu- 
lum whose length is eqnal to the minor axis. 

The fnime ellipse also possesses this curious and useful, property^ 
Tiz. :— That any segment or any £one of the ellipse cut oflf by lines 
parallel to the major axis, whether it be taken near the upper part 
of the minor axis, near the middle, or near the bottom of the same, 
will vibrate in the same time as the whole ellipse^ the point of sus- 
pension being at an extremity of the minor axis. 

26. It is evident from § 1 7, page 228, that pendulums in differ- 
ent latitudes require to be of different lengths, in order that they 
may perform their vibrations in the same time; but besides this 
there is another irregularity in the motion of a pendulum in the same 
place, arising from the different degrees of temperature. Heat ex- 
panding, and cold contracting the rod of the pendulum, certain 
small variations must necessarily follow in the time of its vibration ; 
to remedy which, Tarious methods have been invented for construct- 
ing what are commonly called compensation pendulums^ or atich as 
shall always preserve the same distance between the center of oscil- 
lation and the point of suspension ; and of these we shall describe 
two or three. 

Comf^ensalion pendulums have received different denominations, 

from their form and materials, as the gridiron pendulum^ mercurial "^ 
pendulum, &c. 

27. The gridiron pendulum consists of five rods of steel, and four -m- ,i 
of brass, placed in an alternate order, the middle rod being of steel, ..^ f J, 
by which the pendulum ball is suspended ; these rods of brass andf^ d 
steel are placed in an alternate order, and so connected with eachn^-h 
other at their ends, that while the expansion of the steel rods has s^ a 
tendency to lengthen the pendulum, the expansion of the brass rods .Ezds 
acting upwards tends to shorten it. And thus, when the lengths o ^n^oi 
the brass and steel rods arc duly proportioned, theur expansions an<» mznd 
contractions will exactly balance and correct each other, and so pre^^»-e- 
serve the pendulum invariably of the same length. Sometimes 3, t w^~ 7, 
or 9 foda, are employed in the construction of the gridiron penduKi^v .u- 
lum ; and zinc, silver, and other metals may be used instead of bras=%.^BS8 
and steel. 

28. The mercurial pendulum was invented by Mr. Graham, dt^w^ an 
eminent clockmaker, about the year 1715. Its rod was made ^ of 
brass, and branched towards its lower end, so as to embrace a cyliB^ci Ji- 
dric glass vessel 13 or 14 inches long, and about 2 indiea diamet er- ^^r ; 
which, being filled about 12 inches deep with mercury, forms iJ^B ie 
weight or ball of the pendulum. The height of the mercury in ti^We 
glass being so proportioned to the length of the rod that its expansi^*^^^ 
and contraction exactly balanced the expansion and contraction ^^1^ 
the pendulum rod, and preserved the distance of the center of osc ^■'Z' 
lation from the point of suspension invariably the same. 

This kind of pendulum fell entirely into disuse soon after Graham's 



/ 



CHAP. III.] GYBATION AND ROTATION. 229 

time ; but it has lately been reniclopted with considerable success by 
practical astronomers. A very instractiTe paper on its principles, 
construction, and use, has been published by Mr. F, Bailjfy in vol. i. 
part 2, Memoirs of the Astronomical Society of London; in which 
paper is also contained an extensive and valuable table of the expan- 
sion of different substances by beat. 

29. Reid's compensation pendulum is a recent invention of Mr. 
Adam Beid, of Woolwich, the construction of which is as follows : — 
A N (fig. 189) is a rod of wire, and Z Z a hollow tnbe of zinc, which 
slips ou the wire, being stopped from falling off by a nut N, on which 
it rests ; and on the upper part of this cylinder of zinc rests the 
heavy ball B: now the length of the tube ZZ being so adjusted to 
the length of the rod A N, that the expansions of the two bodies shall 
be equal with equal degrees of temperature; that is, by making the 
length of the zinc tube to that of the wire, as the expansion of tlie wire 
is to tliat of zinc, it is obvious that the ball B will in all cases pre- 
serve the same distance from A; for just so much as it would descend 
by the expansion of the wire downwards, so much will it ascend by 
the expansion of the zinc upwards, and consequently its vibrations 
will in all temperatures be equal in equal times. 

30. Drummond s compensation pendulum was proposed by an artist 
of that name, in Lancashire, more than 70 years ago. A bar of the 
same metal with the rod of the pendulum, and of the same thick- 
ness and length, is placed against the back part of the clock case ; 
from the top of this a piece projects, to which the upper part of the 
pendulum is connected by two fine pliable chains or silken strings, 
which just below pass between two plates of brass whose lower 
edges will alwavs terminate the lengtli of the pendulum at the upper 
end. These plates are supported on a foot fixed to the back of the 
case. This bar rests upon an immoveable base on the lower part of 
the case, and is braced into a proper groove, which admits of no 
motion any way but that of expansion and contraction in length by 
heat and cold. In this construction, since the two bars are of equal 
magnitude and of the same material, their expansions and con trac- 
tions will always be eqnal and in opposite directions ; so that one 
will serve to correct and annihilate the effects of the other. 

CENTER OP OYBATION AND THE PRINCIPLES OP ROTATION. 

1. The center of gyration is that point in which, if all the matter 
contained in a revolving system were collected, the same angular 
velocity would be generated in the same time by a given force acting 
at any plaee as would be generated by the same force acting similarly 
ID tbe body or system itself. 

When the axis of motion passes through the center of gravity, 
tlien is the center called the principal center of gyration. 

2. The distance of the center of gyration from the point of sus- 
pensioD or the axis of motion, is a mean proportional between the 
distances of the centers of osdllation and gravity from the same point 
or axis. 



r" 



230 GYRATION AND ROTATION. [PART 11. 

If 6 equal the distance of tbe center of gravity from the point of 
suspension, O the distance of tbe center of oscillation, and R the 
distance of the center of g\Tation from the same point, then we have 

R = >/go (XVII.) 

3. The distance R of tlie center of gyration, from the center or 
axis of motion, in some of the most useful cases, is given below. 

In a circular wheel of uniform thickness ... R = rad >/ \, 
111 the periphery of a circle revolving about) _^ A / X 

the diameter j v s* 

In the plane of a circle ditto R = | rad. 

In the surface of a sphere ditto R =rad s/ §• 

In a solid sphere ditto R = rad ^/t =^ i^rnearly. 

In a plane ring formed of circles whose radii) /** "*" 

are R, r, revolving about its center j ^\/ 2~ 

In a cone revolving about its vertex R = J ^ 3* ^' "+" J »''• 

In a cone revolving about its axis R = rv^-j^y. 

In a paraboloid R = r ^^ i- 

/ R' + f^ 

In a straight lever whose arms are R and r, R = a / — r ; • 

^ . V srR + r) 

4. If the matter in any gyrating body were actually to be placed 
as if in the center of gyration, it ought either to be disposed in the 
circumference of a circle whose radius is R, or at two points R, R^ 
diametrically opposite, and each at the distance R from the center. 

5. By means of the theory of the center of gyration, and the values 

of R = ^, thence deduced, the phenomena of rotation on a fixed axis ^ 

become connected with those of accelerating forces : for then, if a ,^ 

weight or other moving ])ower P act at a radius r to give rotation to ^y 

a body whose weight equals W, and the distance of whose center of "^ 

gyration from the axis of motion equals ^, we shall have for the -^^-^ 
accelerating force, the expression 

/=P^^ (^v"^> 

and consequently for the space described by the actuating weight or "tk-^di 
power P, in a given time ty we shall have the usual formulse (V.) and f» jd 
(VII.), page 214, 






introducing the above value of/. 

6. In the more complex cases, the distance of the center of gyra -^- 

tion from the axis of motion may best be computed from an experi ^" 

ment. Let motion be given to the system, turning upon a horiatontal^-^ 
axis, by a Mcight P acting by a cord over a pulley or wheel whose^^^ 
radius equals r, fixed upon the same axis, and let s be the space ^^^ 
through which the weight P descends in the time /, the proposed ^^ 
body whose weight is W turning upon the same axis with the wune— ^ 
angular velocity ; then 



i 



CHAP. III.] AXES OP ROTATION. 231 

K=,.^€£f!lz^ (--) 

Example. A body which weighs 100 lbs. turns upon a horizontal 
axis, motion being communicated to it by a weight of 10 lbs. hanging 
from a very light wheel of 1 foot diameter. The weight descends 
2 feet in 3 seconds. Required the distance of the center or circle of 
gyration from the axis of motion. 

" Here, potting y = 32, instead of 32^, we obtain as an approxima- 
tiye result, 

R = ^ /32xl0x9xi-4xl0xi ^ ^.33^3 ,^^^ 
V 4 X 100 

I. When the impulse communicated to a body is in a line passing 
through its center of gravity, all the points of the body move forward 
with the same velocity, and in lines parallel to the direction of the 
impulse communicated. But when the direction of that impulse docs 
not pass through the center of gravity, the body acquires a rotation 
OD an axis, and also a progressive motion, by which its center of 
gravity is carried forward in the same straight line, and with the 
same velocity, as if the direction of the impdse had passed through 
the center of gravity. 

The progressive and rotatory motion are independent of one 
another, each being the same as if the other had no existence. 

8. When a body revolves on an axis, and a force is impressed, 
tending to make it revolve on some other, it will not revolve on 
either, but on a line in the same plane with them, dividing the angle 
which they contain, so that the sines of the parts are in the inverse 
ratio of the angular velocities with which the body would have 
revolved about the said axes separately. 

9. A body may begin to revolve on any line as an axis that passes 
through its center of gravity, but it will not continue to revolve per- 
manently about that axis, unless the opposite rotatory forces exactly 
balance one another. 

This admits of a simple experimental illustration. Suspend a thin 
circular plate of wood or metal by a cord tied to its edge, from a hook 
to which a rapid rotation can be given. The plate will at first turn 
upon an axis which is in the continuation of the cord of rotation ; but 
as the velocity augments, the plane will soon quit that axis, and 
revolve permanently upon a vertical axis passing through its center of 
gravity, itself having assumed a horizontal position. 

The same will happen if a ring be suspended, and receive rotation 
in like manner. And if a flexible chain of small links be united at 
its two ends, tied to a cord and receive rotation, it will soon adjust 
itself so as to form a ring, and spin round in a horizontal plane. 

Also, if a flattened spheroid be suspended from any point, how- 
ever remote from its minor axis, and have a rapid rotation •:ivcn it, it 
nill ultimately turn upon its shorter axis ])08itod vcrticully. This 
evidently serves to confirm the motion of the earth upon its shorter 
axis. 



232 CENTHAL FORCES. [PABT II 

10. In every body, however irregular, tbere are three axes of per 
mancnt rotation, at right angles to one another. These are callei 
the principal axes of rotation; and they have this remarkable pro- 
perty, that the raomentam of inertia with regard to any of them i 
either a maximum or a minimum. 

CENTRAL FORCES. 

1 . Centripetal force is a force which tends constantly to solicit o 
to impel a body towards a certain fixed point or center. 

2. Centrifugal force is that by which it would recede from such 
center, were it not prevented by the centripetal force. 

3. These two forces are, jointly, called central forces. 

4. If W denote the weight of a body moving in a circle* whoa 
radius equals r, with the velocity t?, its centrifugal force scy will hm 

f'"-^ ■ ■ ■■- (-•) 

where g equals the force of gravity. 

If t equal the time of one revolution, and ir = 31 41 59, then 

/=i^=-!^w («,., 

5. When a body describes a eirde by means of a force directed 
its center, its actual velocity is every where equal to that which, 
would acquire in falling by the same uniform force through half fc 
radius. 

6. This velocity is the same as that which a second body woi^ 
acquire by falling through half the radius, whilst the first described 
portion of the circumfereuce equal to the whole radius. 

7. In equal circles the forces are as the squares of the times » 
versely. 

8. If the times are equal, the velocities are as the radii, and C 
forces are also as the radii. 

9. In general, the. forces nre as the distances or radii of the drcf 
directly, and the squares of the times inversely. 

1 0. The squares of the times are as the distances directly, and tl 
forces inversely. 

11. Hence, if the forces are inversely as the squares of the di 
tances, the squares of the times are as the cubes of the distance 
That is, 

if F :/ :: rf« : D«, then T- : f ;: D' : d\ . . (XXII.) 

12. The right line that joins a revolving body and its center of a 
traction, called the radius vector^ always describes eqnal arcaa in eqa 
times, and the velocity of the body is inversely as the perpendical 
drawn from the center of attraction to the tangent of the t^nrve 
the place of the revolving body. 

13. If a body revolve in an elliptic orbit by a force directed to o 
of the foci, the force is inversely as the square of the distance: « 



CHAP. III.] CENTRAL FORCES. 233 

the mean distances and the periodic times have the same relation as 
in Art. 11. This eomprtkenda the case of the planetary motions. 

14. If the force which retains a hody in a curve increase in tlie 
simple ratio as the distance increases, the body will still describe an 
ellipse; but the force will in this case be directed to the center of the 
ellipse; and the body in each revolution will twice approach towards 
it, and again twice recede from that point. 

15. On the principles of central forces depends the operation of the 
conical pendulum applied as a governor or regulator to steam engines, 
water mills, &c. 

This contrivance will be readily comprehended from fig. 1 90, where 
A a is a vertical shaft capable of turning freely upon the sole a. C D, 
e F, are two bars which move freely upon the center C, and carry at 
their lower extremities two equal weights, P, Q; the bars CD, C F, 
are united, by a proper articulation, to the bars O, H, which latter arc 
attached to a nng 1, capable of sliding up and down the vertical 
shaft A a. When this shaft and connected apparatus are made to 
revolve, in virtue of the centrifugal force, the balls P Q fly out more 
and more from A a, as the rotatory velocity increases: if, on the con- 
trary, the rotatory velocity slackens, the balls descend and approach 
A a. The ring I ascends in the former case, descends in the latter: 
and a lever connected with I may be made to reguhUe the energy of 
the moving power. Thus, in the steam engine, the ring may be made 
to act on the valve by which the steam is admitted into the cylinder; 
to augment its opening when the motion is slackening, and recipro- 
cally diminish it when the motion is accelerated. 

The construction is often so modified, that the fiying out of the 
balls causes the ring I to be depressed,, and vice versd; but the gene- 
ral principle is the same. 

Here, if the vertical distance of P or Q below C, be denoted by c/, 
the time of one rotation of the regulator by ^, and 3- 14 1593 by ^, 
the theory of central forces gives 

/ = 2 ^ /v/s^ = ^'^^^^^ >/ rf . . (XXIII.) 

Hence, the periodic time varies as the square root of the altitude 
of the conic pendulum, let the radius of the base be what it may. 
Also, when ICQ = ICP=s 45°, the centrifugal force of each ball 
is equal to its weight. 

16. As the practical utility of the conical pendulum depends in a 
great degree upon iu sensibility, or the change which must take place 
m its Telocity before it will move the ring 1 to the required extent, 
we subjoin the following formula for determining the weight of the 
balls P and Q, for any degree of sensibility which may be required. 

LetW equal the weight of both the balls, P the power required to 
move the valve (or produce whatever effect may be required) when 
applied to the ring in the direction of the spindle, J^, the number of 
revolutions which the governor is intended to make per second, N^ 
the number of revolutions which the same must make to move the 



234 iNQriitiEs cuNNfccTiiii WITH [part II. 

ring I, </ as before the vertical distance from the plane of the balls 
P and Q to the point C, b tlie distance CD or C F, / the distance 
C P, a the distance of the point D from the axis of the spindle, c the 
vertical distance from the plane joining the points D and F from the 
point I, and r the radius of the circle described by the balls, or their 
distance from the axis of the spindle ; then 

'-iJ^is? ^^^'^•> 

If the distance C D is made equal to D I, so that the four rods form a 
])aralleIogram, this last formula becomes 

^^' = i-.-.73r</?(N?-3]^) • • • (^^^'-^ 

17. In the foregoing formulte the center of oscillation of the balls 
and system of rods is assumed to coincide with the center of the 
balls, an assumption not strictly correct, although sufficiently so for all 
practical purposes. Should, however, greater accuracy be required, 
the true center of oscillation having been found, the following sub- 
stitutions must be made, when the preceding formulae will give an ac- 
curate result ; viz., for /, the distance from C to the center of oscilla- 
tion, and for r the distance of the center of oscillation from the ver- 
tical axis of the spindle. 

INQUIRIES CONNECTED WITH ROTATION AND CENTRAL FORCES. 

1. Suppose the diameter of a grindstone to be 44 inches, and iisa 
weight half a ton ; suppose also that it makes 386 revolutions in tm 
minute. What will be the centrifugal force, or its tendency to burst ^ 

44 386 

Here the velocity = 75 X 31416 x --;--- = 74106; 

then by substituting these values in formula (XX.) we have 

^ 74-1062 X -5 

/= - 7 -- = 46-8 tons. 

•^ 32 X -tj 

the measure of the required tendency. 

2. If a fly wheel 12 feet diameter, and 3 tons in weight, revolivi 
in 8 seconds : and another of the same weight revolves in 6 seconds 
what must be the diameter of the last, when their centrifugal force 
the same ? 

By formula (XXI.) F : / : : ~ : ^. Therefore, since F is = ,,^ 

D d , Vt' 12 X 36 ^, ^ 

2^ = ;;}> ^»' « = -^P" = J54— = ^J ^^^^ ^^^ answer. 



CHAP. III.] ROTATION AND CENTRAL FORCES. 235 

3. If a fly of 12 feet diameter revolve in 8 seconds, and another 
of the same diameter in 6 seconds : what is the ratio of their weights 
when their central forces are equal ? 

By § 7, page 232, the forces arc as the squares of the times in- 
versely when the weights are equal : therefore, when the weights are 
unequal, tbey must be directly as the squares of the times, that the 
central forces may be equal. 

Hence » : W :: 36 : 64 :: 1 : 1^ 

That is, the weight of the more rapidly to that of the more slowly 
revolving fly, must be as 1 to 1^, in the case proposed. 

4. If a fly 2 tons weight and 1 G feet diameter, is suflicient to regu- 
late an engine when it revolves in 4 seconds; what must be the 
weight of another fly of 12 feet diameter revolving in 2 seconds, so 
that it may have the same power upon the engine ? 

Hertf, by § 9, page 232, we must have — — = -^ ; therefore 

40cwt. X 16 X 4 160 ,^, 1^ . , 

s - - ^ ^ = -;« = 134 cwt., the weight 

12 X 16 12 3 » 6 

of tbe smaller fly. 

NoUj'-^K fly should always be made to move rapidly. If it be 
intended for a mere Regulator, it should be near the fimt mover. If 
it be intended to accumulate force in tbe workittg pointy it must not 
be far separated from it. 

5. Given the radius R of a wheel, and the radius r of its axle, the 
weigbt of both, tp, and the distance of the center of gyration from the 
axis of motion, ^ ; also a given power P acting at the circumference 
of the wheel ; to find the weight W raised by a cord folding about 
tbe axle, so that its momentum shall be a maximum. Here 

^_ s/ (R^ P* -f 2 R*Pg* w - h e^y* - f P»Rrg* -f P*RV)— R'P— g*ig 

Cor. 1. When R =s r, as in the case of the single fixed pulley, 
(ben 

W= >/(2P»R' + 2RPf*»+^»« -h P»Re») — ^»— P. 

R R" 

Cor. 2. When the pulley is a cylinder of uniform matter e* = J R% 
^•^d the expression becomes 

W = VlR' (2 P« -h 4 P » + 4 tP*)} — i w — P. 

6. Let a given power P be applied to the circumference of a wheel, 
^'liose radius equals R, to raise a weight W at its axle, whose radius 
^^ r, it is required to find the ratio of R and r when W is raised with the 

^^^eatest momentum ; the characters W and ^ denoting the same as in 
^^^c last proposition. 



236 COLLISION OF BODIES. [PART I 

Here r = ; — r • 

P (y -r W) 

Cor. Wlien the inertia of the machine is eTanesoent, with retp© 
to that of P -I- W, then is r = R ^ / (\ + ^) — 1- 

7. In any machine whose motion accelerates, the weight ^-ill I 
moved with the greatest velocity when the velocity of the power is 1 

that of the weight as 1 + P a / (^ + T^ ) ia to 1 ; the inertia i 

the machine heing disregarded. 

8. If in any machine whose motion accelerates, the deseent of oc 
weight causes another to ascend, and the descending weight be give- 
the operation being supposed continually repeated, the effect vrill 1 
greatest in a given time when the ascending weight is to the du 
scending weight, as 1 to 1*618, in the case of equal heights; and 
other cases when it is to the exact counterpoise in a ratio which^ 
always between I to 1| and 1 to 2. 

9. The following general proposition with regard to rotatory nz 
lion will be of use in the more recondite cases. 

If a system of bodies be connected together and supported at «■ 
point which is not the center of gravity, and then left to descend 
that part of their weight which is not supported, 2g multiplied \m 
the sum of all the products of each body into the space it has p^- 
pendicularly descended, will be equal to the sum of all the prodiB 
of each body into the square of its velocity. 



Sect. IV. Percussion or Collision of Bodies in motion. 

1. In the ordinary theory of percussion, or collision, bodies - 
regarded as either hard^ sojly or elastic, A hard body is that wh— 
parts do not yield to any stroke or percussion, but retains its fig* 
unaltered. A soft body is that whose parts yield to any stroke 
impression, without restoring themselves again, the shape of the be: 
remaining altered. An elastic body is that whose parts yield to m 
stroke, but presently restore themselves again, so that the body 
gains the same figure as before the stroke. When bodies which he 
been subjected to a stroke or pressure return only in part to i\m 
original form, the elasticity is then imperfect: but if they rest* 
themselves entirely to their primitive shape, and employ just as mu. 
time in the restoration as was occupied m the compression, then 
the elasticity perfect. 

It has been customary to treat only of the collision of bodies p^ 
fectly hard or perfectly elastic : but as there do not exint in nat* 
any bodies (which we know) of either the one or the other of th 
kinds, the usual theories are but of little service in practical i* 



CHAP. III.] COLLISION OP B0DTK8. 237 

cbanics, except as tliey may suggest an extension to tbe actual cir- 
cumstances of nature and art. 

2. Tbe general principle for determining tbe motions of bodies 
from percussion, and wbicb belongs equally to both elastic and 
non-elastic bodies, is this : viz. that there exists in the bodies the 
same momentum, estimated in any one and the same direction, both 
before the stroke and after it. And this principle is the immediate 
result of the law of nature or motion, that reaction is equal to ac- 
tion, and in a contrary direction ; from whence it happens, that what- 
ever motion is communicated to one body by the action of another, 
exactly the same motion does this latter lose in the same direction, 
or exactly the same does the former communicate to the latter in tlie 
contrary direction. 

From this general principle too it results, that no alteration takes 
place in the common center of gravity of bodies by their actions 
upon one another ; but that the said common center of gravity per- 
severes in the same state, whether of rest or of uniform motion, both 
before and after the impact. 

3. If the impact of two perfectly hard bodies be direct, they will, 
after impact, ekher remain at rest, or move on uniformly together 
with different velocities, according to the circumstances under which 
they met. 

Let B and h represent two perfectly hard bodies, and let the velo- 
city of B be represented by V, and that of b by r, which may be 
taken either positive or negative, according as h moves in the same 
direction as B, or contrary to that direction, and it will be zero when 
b is at rest. This notation being understood, all the circumstances 
of the motions of the two bodies, after collision, will be expressed 
by the formula : 

, . BV±bv 

velocity = — 

B -f- 

irliich being acoommodated to the three circumstances under which 
9 may enter, become 

„ V , , BV -f 5t? / when both bodies moved in 

(I.) velocity = p^^ \ the same direction 

,,^ . , . BV — bv i when the bodies moved in 

(II.) velocity = p^^ [ contmry directions 

,^ ^ , . B 4- ft ( when the body b was at 

(HI.) velocity =:^^ I rest. 

Tbese formulsB arise fi^m the supposition of the bodies being per- 
fecOj hard, and consequently that the two after impact move on uni- 
formly together as one mass. In cases of perfectly elastic bodies, 
other foraulflB have place which express the motion of each body 
Mparately; as in the following proposition. 

4. If the impact of two perfectly elastic bodies be direct, theii- 
relative velocities will be the same both before and aft^r impact, or 



238 COLLISION OF BODIES. [pAHT 

they will recede from eau^ other with the same velocity with whi- 
they met; that is, they will be equally distant, in equal times, bo^ 
before and after their collision, althou^ the absolute velocity of etu^^ 
may be changed. The circumstances attending this change of m.^ 
tion in the two bodies, using the above notation, are expr^eed in t~l^ 
two following formulae : 

^~X — ^— = velocity of B . . . (IV.) 

2BV + (B-6)v , . ^, 

^ ^— = velocity of 6 . . . (V.) 

which needs no modification, when the motion of & is in the s^^ 
direction with that of B. 

5. In the other case of b's motion, the general formulae become^- 

^^A!Ll_(Ezl*)^ = ,elocity«fB. (VI.) 

2_LL^J!L:ii)^=velockyofft . (VII.) 

when b moves in a contrary direction to that of B, which arises fr€>i 
taking v negative. And 

(B — b)y 

^ -— = velocity of B ( VIII.) 

n n ^ 

- - - = velocity of ^ (IX.) 

when b was at rest before impact, that is, when v = 0. 

G. If a perfectly hard body B, (fig. 191,) impinge obliquely uf?^ 
a perfectly hard and immoveable plane A D, it will after coUi^'^ 
move along the plane in the direction C A. 

And its velocity before impact 
Is to its velocity after impact 
As radius 
Is to the cosine of the angle BCD. 

But if the body be elastic it will rebound from the plane in t 
direction C £, with the same velocity, and at the same angle wi - 
which it met it, that is, the angle ACE will be equal to the an^ 
BCD. 

7. The force with which a body impinging obliquely strikes 
plane, is to the same if it had acted perpendicularly, as the sine ^ 
the angle (B C D) of incidence, is to radius. 

8. In the case of direct impact, if B be the striking body, b tH 
body struck, V and v their respective velocities before impact, V, ax^ 
«, their velocities afterwards; then the two following are genei^ 
formulce: viz. 



CHAP. III.] ON THK MECHANICAL lOWFRS. 23.9 



' v- 
r, = r -f n 



In these, if « = 1, they serve for non-elastic bodies ; if « = 2, for 
bodies perfectly elastic. If the bodies be imperfectly elastic, n has 
some intermediate Talue. 

When the body struck* is at rest, the preceding equations become 

V. = V - -!Lll (XII.) 

» B+b ^ ' 

r,=!^ (XIII.) 

n = ^._(« +i) (XIV.) 

BV ^ ' 

from which the value of n may be determined experimentally. 

9. In the usual apparatus for experiments on Collision, balls of 
different sizes and of various substances are hung from different 
points of suspension on a horizontal bar. MAN (fig. 192) is an arc 
of a circle whose center is S ; and its graduations, 1, 2,3, 4, 5, &c., 
indicate the lengths of chord lines measured from Uie lowest point D. 
Any ball, therefore, as P, may be drawn from the vertical, and made 
to strike another ball hanging at the lowest point, with any assigned 
velocities, the height to which the ball struck ascends on the side 
A M furnishing a measure of its velocity ; and from that the value of 
n may be found from the last equation. Balls not required in an in- 
diridoal experiment, may be put behind the frame as shown at A 
and B. 

The cup C may be attached to a cord, and carry a ball of clay, &c. 
when required. 

Example, Suppose that a ball weighing 4 ounces strikes another 
ball of the same substance weighing 3 ounces, with a velocity of ] 0, 
^d communicates to it a velocity of 8| : what, in that case, will be 
^he value of n ? 

«eren= ' p^ = ^-^^^ = ^j^- = 144375 the index of 
the degree of elasticity; perfect elasticity being indicated by 2.. 



Sect. V. On the Mechanical Powers, 

1. The most complicated machinery is nothing more than a com- 
bination, or constant repetition, of a few simple mechanical expe- 
dients for modifying and changing the direction of the several forces 



240 MECHANICAL POWERS. [PART II. 

or pressures, transmitted by them through the machine. A certain 
pressure or force, being communicated to what is termed the first 
or prime mover of the machinery, is thus transmitted through every 
part of the machinery, being regulated and modified in such a man- 
ner as to produce the effect required, at what is termed the reork- 
ing point. Thus, by means of the machine termed a crane, a man, by 
applying the pressure of his hand (amounting to perhaps about 30lb6.) 
to the handle of the crane, is enabled, through the intervention of 
the machinery, to raise an enormous weight, as, for instance, say 
12,000 lbs., which, without some such expedient, would defy all his 
efforts to move. 

2. These elementary parts of which more complicated machinery 
is composed, have been called the mechanical powers^ a term which 
is liable to lead to a misapprehension of the effects which they are 
really capable of producing. For, ^continuing the use of the previous 
illustration,) although the man, through the instrumentality of the 
crane, by exerting a ^rce of SOlbs., is enabled to lift 12,000lbs., or four 
hundred times the amount, it will be found that his hand will move 
four hundred times as fast as the weight, and, therefore, what advan- 
tage he may appear to eain in the weight which he is enabled to •« 
lift, he really loses again by the length of time which it takes him to ^ 
raise it to any given height ; and which would be found by experi- — 
ment to be just what he would require to raise separately four hnn- — 
dred weights, each of dOlbs., to the same height. 

3. The nnmber of the mechanical powers is usually reckoned to ^ 
be six : viz. the lever ^ the wheel and a^rfe, the pttUey, the indined^^ 
planey the wedge, and the screw. 

4. In treating of these machines, we use the word power^ to denote ^s 
the force which is supposed to be exerted at the origin of the machine, ^.i^ 
and the M'ord weight to denote the effect which that force produces^^ 
at the working point of the machhie : and we shall express them by "^ 
the letters P and \V respectively. 

LEVERS. 

1 . A lever is an inflexible bar, whether straight or bent, and sup^ 
posed capable of turning upon a fixed, unyielding point, called aful^ 
crum. There are three kinds of levers. 

2. When i\\t fidcrum is between the power and the weight, as iia 
fig. 193, the lever is said to be of \\\q first kind, 

3. When the weight is between the power and the fulcrum, as in 
fig. 194, the lever is of the second kind. 

4. When the power is between the weight and the fulcrum, as in 
fig. 1 95, the lever is of the third kind. 

The bent lever, as employed in the operation of drawing a nail with 
a hammer, is sometimes considered as a fourth kind, but is really a 
lever of the first kind. 

.5. In all these coses, where there is an equilibrium, it is indicated 
by this general property, that the product of the weight into the dis- 
tance at which ft acts, is equal to the product of the power into the 



CBAf.lu,] MECHANICAL POWERS. 241 

distance at which it acts : tke distances being estimated in directione 
P^'yndieidar to those in which the weight and power act respectively. 
3T>|», ID each of the three preceding figurcR, 

P. AF = W. BF, 

*• the power and weight are reciprocally as the distances at which 
^cyact. 

If, in fig. 193y for example, the arm A F were 4 times F B, 4 Ihs. 
ftngfng at B would he halanced hy 1 Ih. at A ; and if A F were 5 
Enes FB, 1 Ih. at A would halance 5 Ihs. at B; and so on. 

6. If several weights hang upon a lever, some on one side of the 
ilcnun, some on the other, then there will be an equilibrium, when 
« sum of the products of the weights into their respective distances 
1 one ade, is equal to the several products of weights and distances 
la the other side. 

Tor, the product of the weight into its distance from the fulcrum 
\ is the same as its trioment about the point F, and therefore the 
OTegmng proposition is a direct consequence of the principle of the 
e«)Yiality of the moments of any forces in equilibrium, about any 
fixed point, as explained at § 5, page 189. 

7. When the weight of the lever is to be taken into the account, 

F^oceed just as though it were a separate weight suspended at the 

^We of its center of gravity. 

8. When two, three, or more levers act one upon another in sue • 
lesion, then the entire mechanical advantage which they afford, is 
foond by taking, not the eum^ but the product of their separate ad- 
stages. Thus, if the arms of three levers, acting thus in con- 
P^on, are asStol, 4to1, and 5 to 1, then the joint advantage 
» that of 3 X 4 X 5 to 1, or 60 to 1 : so that 1 lb. would, through 
^ intervention, balance 60. 

9. Id the first kind of lever the pressure upon the fulcrum = P 
t W; in the other two it is = P '^ W. 

10. Upon the foregoine principles depends the nature of scales 

■Bd beams for weighing all bodies. For, if the distances be equal, 

tko will the weights be equal also ; which gives the construction of 

tlie common scales. And the Roman statera, or steel-yard, is also a 

Uifer^ but of unequal arms or distances, so contrived that one weight 

anJy may serve to weigh a great many, by sliding it backwards and 

Avwsrds to different distances upon the longer arm of the lever. In 

tfce common halanc-e^ or scales, if the weight of an article when 

•soertained in one scale is not the same as its weight in the other, 

Mr square root of the product of those two weights will give the true 

metgit. 

]]. From nnmeroos examples of the power and use of the lever, 
we which shows its manner of application in the printing-presses of 
dbe late Earl Stanhope may be advantageously introduced. 



242 



MECHANICAL POVfBBS. 



[part II. 




In the adjoining figure, let A BCD 
be the general frame of the press, 
connected bv the cross pieces N O, 
DC. E is a center connected with 
the frame by the bars E N, E R, E O. 
To this center arc fixed a bar KL, 
and a lever E F, to which the hand is 
applied when the press is nsed. 

Tiiere are also several other pieces 
connected by joints at N, 6, 1, K, L, 
M, O, H, which are so adjusted to 
each other, that when the hand is applied to the lever EF at F, by ^\^^ 

f)ressing it downwards KL is brought into a horizontal line or paral- ^JTg-^ 
el to O U or D C, in which situation N I O, O M H, also form each omm j 
straight line. It is evident that the nearer these different pieces, t^"i a^ | 
above mentioned, are to a straight line the greater is the lever EF. ~=s^ f 
in proportion to the perpendicular KS at the other end of the lever ^^ -^^ 
EK, formed by a perpendicular from K falling on F£ prodocedE:^ •?(/. 
Consequently a small force applied at F will be sufficient to produce ^g 

very great effect at K, when I K, K E are nearly in a straight line, an .^i^ajif 
so on, for the other pieces above mentioned. 

Hence the force applied by hand at F must be very considerable ^/e 
in forcing down OH, which slides on iron cylindrical bars, or ? in 
pressing any substance placed in the aperture PQ, between the be- ^^ 
or plate and the frame DC. 

This contrivance is now often introduced into mechanism, und er 

the name of the toggle ^ or knee-Joint, 

WHEEL AND AXLE. 

1. The nature of this machine is suggested by its name. To it 
may be referred all turning or wheel-machines containing wheels of 
different radii ; as well-rollers and handles, cranes, capstans, \iir^ ^' 
lasses, &c. 

2. In the wheel and axle the mechanical property is the sam^ u 
in the lever: viz. P . AC = W . BC (fig 196); and the reaaoim i» 
evident, because the wheel and axle is only a kind of perpetrnv/ 
lever. 

3. When a series of wheels and axles act upon each other, so ^ 
to transmit and accumulate a mechanical advantage, whether the 
communication be by means of cords and belts, or of teeth nod 
pinions, the weight will be to the power, not as the mm, but as tbe 
continual product of the radii of the wheels to the continual product 
of the radii of the axles. Thus, if the radii of the axles, a,ft,e,A<i 
(fig. 197,) be each three inches, while the radii of tlie whedif 
A, B, C, D, £, be 8, 6, 9, 10, and 12 inches respectively: tbtn 
W:P::9xex9xlO x 12 : 3 X 3 X 3 X 3 X 3 :: 240:1. i 
computation, however, in which the effect of friction is disr^arded. 



PULLEY. 

palley is a small wheel, commonly made of wood or brass, 
of turning upon an iron axis passing through its center, and 

a block, and the use of which is, that by means of a cord 
ound its circumference, we are enabled to alter the direction 
pce in any way that we choose. The pulley is either single 
purpose, or combined with others to obtain a mechanical ad- 
. It is also either fixed or moveable, according as it is fixed 
ilacc, or moves up and down with the weight or power*. 

a power sustain a weight by means of a fixe<l pulley, as in 
, the power and weight are equal. 

lere are several different methods of combining pulleys to- 
or forming as they are termed separate systems of pulleys, for 
g 4 mechanical advantage. ^ 

first system is shown in fig. 199, in which there are two 
f pulleys, the upper fixed, and the lower attached to the 
and rising with it ; only one continuous rope is employed, 
lay be attached to either block, and passed successively round 
dley. In this system, the weight sustained at W, is to the 
sustaining it at P, as the number of ropes engaged between 

blocks, is to 1 ; thus in fig. 199 a, the mechanical advantage, 
I, and in fig. 199 5, it is 4 to 1. 

be second system is when there are as many separate ropes 
! are pulleys, each rope being attached to a fixed point at one 
I, passing under one of the pulleys, has its other end attached 
lock of the next pulley above it. This system is represented 



•elul eombinaUon of the wheel and axle, a fixed and a moveable pulley 
ted in the marginal dia- 

rhe loadf as of stones or >4b 

» Imild a wall, is raited from F ^^ 

tkufs a rope BP Lis fixed UL 



*_ - I U I* I — 





CHAP. III.] MECHANICAL POWERS. 245 

Tkas, suppose the angle A B H was 30°, D B I 6*0% and con* 
seqiiently ABD 90°: since tlic natural sines of 90% 60°, and 30°, 
are 1, -866, and -5 respectively, or nearly as 100, 86*6, and 50; if 
the heavy body weigh 100 lbs., the pressure upon AB would be 
86'f> 1bs.i and upon BD 50 lbs. 

This proposition is of very extensive utility, comprehending the 
pressure of arches on their piers, of buttresses against walls, or upon 
the ground, &c., because the circumstance of one of the pianos 
becoming either horizontal, or vertical, will not affect the general 
relation above exhibited. 

WEDOB. 

1. A wedge is a triangular prism, or a solid conceived 
to be generated by the motion of a plane triangle parallel 
to itself upon a straight line which passes through one of 
its angular points. The wedge is called isosc^es^ rect- 
angtUar, or 9calene^ according as the generating triangle 
IS isosceles, right-angled, or scalene. It is very fre- 
quently used in cleaving wood, as represented in the figure, and often 
in raising great weights. 

2. When a resistmg body is sustained against the face of a wedge, 
by a force acting at right angles to its direction ; in the case of equi- 
Hbrium, the power is to the resistance as the sine of the semi-angle 
of the wedge, is to the sine of the angle which the direction of the 
resistance makes with the face of the wedge; and the sustaining 
force will be as the cosine of the latter angle. 

3. When the resistance is made against the face of a wedge by a 
body which is not sustained, but will adhere to the ])lace to which it 
is applied without sliding, the power is to the resistance, in the ease 
of equilibrium, as the cosine of the difference between the semi-angle 
of the wedge and the angle which the direction of the resistance 
makes with the face of the wedge, is to radius. 

4. When the resisting body is neither sustained nor adheres to the 
point to which it is applied, but slides freely along the face of the 
wedge, the jwwer is to the resistance as the product of the sines of 
the semi-angle of the wedge and the angle in which the resistance 
ia inclined to its face is to the square of radius. 

SCREW. 

1. The screw is a spiral thread or groove cut round a cylinder, and 
erery where making the same angle with the length of it. So that 
if the surface of the cylinder, with this spiral thread on it, were un-» 
^Ided or developed into a plane, the spiral thread would form a 
^rmight inclined plane, whose length would be to its height^ as the 
^'rcnmference of the cylinder is to the distance between two threads 
^f the screw : as is evident by considering that, in making one 
'^und, the spiral rises along the cylinder the distance between the 
^^o threads. 

2, The energy of a power applied to turn a screw round, is to 
^W force witli which it presses upward or downward (setting aside 



246 



MECHANICAL POWERS. 



the friction), as the distance between two threads is to th 
ference where the power is applied : viz., as the circumferei 
is to the distance BI (fig. 205). 

3. The endless screw^ or perpetual serew^ is one which 
and tnms a toothed wheel DF (fig. 206), without a concave 
screw; being so called because it may be turned for crei 
coming to an end. From the diagram it is evident that 
screw turns once round, the wheel only advances the d 
one tooth. 

4. If the power applied to the lever, or handle of a 
screw, A B, be to the weight, in a ratio compounded of the 
of the axis of the wheel, EH, to the periphery describe 
power in turning the handle, and of the revolutions of the i 
to the revolutions of the screw CB, the power will bn 
weight. Hence, 

5. As the motion of the wheel is very slow, a small p 
raise a very great weight by means of an endless screw. A 
fore the chief use of such a screw is, either where a great 
to be raised through a little space, or where only a si 
motion is wanted. For which reason it is very 8er\nceabh 
and watches. 

The screw is of admirable use in the mechanism of mi 
and in the adjustments of astronomical and other instrun 
refined construction. 

6. The mechanical advantage of a compound machini 
determined by analyzing its parts, finding the mechanical 
of each part severally, and then blending or compound]: 
ratios*. Thus, if m to 1, n to 1, r to 1, and e to 1, 
separate advantages; then mnr s io 1, will measure the ad 
the system. 

* The marginal representation of a 
common construction of a crane to raise 
heavy loads, will serve to illustrate this. 
By human energy at the handle a, the 
pinion h is turned; that gives motion to 
the wheel W, round whose axle, r, a cord 
is coiled; which cord passes over the fixed 
pulley, d^ and thence over the fixed triple 
block, B, and the moveable triple block, P, 
below which the load, L, hangs. Now, 
if the radius of the handle be 6 times that 
of the pinion, the radius of the wheel W 
10 times that of iu axle, and a power 
equivalent to 30 lbs. be exerted at a; 
then, since a triple moveable pulley gives 
a mechanical advantage of 6 to 1, we 
shall have 

30x6xl0x6» 10800 lbs. 
and such would be the load, L. that might be raised by a powe 
applied at a, were it not for the loss occasioned by friction. 




HI.] 



MBCHANICAL POWERS. 



247 



making such a calculation, the subjoined table, exhibiting at one 
the ratio of the power to the weight, in all the simple mechani- 
wers, will be of service. 



ipcioo oi Power* 



and axle . 

id wheels 

§:— 

tem (fig. 199) 

lo. (fig. aoo) 

to. (fig. 201) 
9d plane 



Ratio of 
P : W. 



■■^7 



i.S. 

r 
n 



1 :n 
1 :^- 
1 : 2--1 



Symbol 



L equals the leverage 
of the power, / that 
of the weight. 

I Resrad. of the wheel, 
{ r that of the axle. 
\ N •» No. of teeth in 
I the wheel, n the No. 
[ in the pinion, 

) n equals the No. of 
> pulleys, both fixed 
) and moveable. 

IX =» the length of the 
plane, h its height. 

Cx — the length of the 
J side of the wedge, b 
i the thickness of its 
Chack. 
^ e<a the drcumferenoe 
V of the circle described 
J by the power, and d 
/the vertical disUnce 
^between two threads. 



ObMrvation. 



/ In all the cases both 
, the friction and weight 
j of the machines them. 
^ selves are neglected, 



The strings are all 

supposed to be pa- 

raUel. 

When the power acts 

in a direction parallel 

to the plane. 

f When the resistance 

J inperpendiculartothe 

y side of the wedge, the 

(, wedge being single. 



FABT II 



CEAF. IT. 
5«rT. L Gttfffrm DrfimiTm m M . 



Er3^!»T^r:'3 cniL7nHe» nie -Sacszae of sLe pumme and the 
~ rivsL ic itin-^iassx f iba&. a» v^iicr, aerrair, &c^ mnd that of 

2. A ffwfKt jtmmi s a Vocj vintfe parts are rerr miDOte, vieldiag 
to acT f!:«Te lAzrewvc XT«aK h . Xiov«icr saail), uid bj ao jielding 



srre Mifff^mr t ira l ideas of a flaid bodr, bj com j 

: h %o a beap of a^ ; b«t die xBiTkwiabuitj of ^^^^ flaiditv b? ii^p^i 
aaj \jzA of 3Kc£u3caJ cosauaiitaoii. vill apfwar br eonsidenii^ twci ■ wi 
9i tibe cRsaMcaaen ttpcrwrr to coofdtate a fluid bodj : I . That^i^=t 
tbe parts. iMKvidataDdin* aar compmaoii. maj be mored in 
tioa to eadk oiLer, aiih tbe smallest coooeiTabie foree, or will gire 
DO mnuMt retu^mmct to mocioo wiibio the mass in anj directioD. 
2. That the parts shall sravitate to each ocber, wfaerebr there is 
eoostant teodencT to amnse tbemselTes about a common center, i 
form a j^pKerical bodv ; which, as the parts do not resist motion, i^^^ 
easilj effected in soiali bodies. Henee the appearance of drops^^ 
alurajs takes p!ace when a jlmiJ is in proper circamstances. It i^^^ 
obrions that a body of sand can bj no means conform to thiMJiii^ 
drciraistances. 

Differeot fluids hare different degrees of flaiditr, according to th^^ 
^KiIitT with which the particles maj be mored amongst each other. — 
Water and mercary are classed anumg the most perfect floids— ^ 
Many floids hare a rcrr sensible degree of tenacity, and are therefor^^' 
called viscous or imperfect fluids. 

3. Fluids may be diTided into compressible and ineompressMe^ otf^ 
elastic and non-elastic fluids. A compressible or elastic fluid is on^ 
whose apparent magnitude is diminished as the pressure upon it vm 
increaseil, and increued by a diminution of pressure. Such is air.^ 
and the different Tapours. An incompressible or non-dastie flui<3 
(called also a liquid) is one whose dimenuons are not sensibl/ 
affected by any augmentation of pressure. Water, mercury, oil, &c ^ 
are generally ranged under this class. 

It has been of late years proposed to limit the application of the 
term fluidi to those which are ekutiCy and to apply the word liquid 
to such as are non-elastic. 



CHAP. IV.] PKKSSURB OP FLUIDS. 249 

4. The specific gravity or density of any solid or flaid body, is the 
absolute weight of a known volume of that substance ; namely, of that 
volume which we take for unity in measuring the capacities of bodies. 



Sbct. II. Pressure and Equilibrium of Non-elastic Fluids. 

1. Fluids press equally in all directions^ upwards, downwards, 
aslant, or laterally. 

This constitutes one essential difference between fluids and solids, 
solids pressing only downwards, or in the direction of gravity. 

2. The upper surface of a gravitating fluid at rest is horizontal. 

3. The pressure of a fluid on every particle of the vessel containing 
it, or of any oUier surface, real or imaginary, in contact with it, is 
equal to the weight of a column of the fluid, whose base is equal to 
tliat particle, and whose height is equal to its depth below the upper 
BDifaoe of the fluid. 

4. If, therefore, any portion of the upper part of a fluid be 
Replaced by a part of the vessel, the pressure against this from below 
niil be the same which before supported the weight of the fluid 

i^emoTed, and every part remaining in equilibrium, the pressure on 
the bottom will be the same as it would be if the vessel were a prism 
Or a cylinder. 

5. Hence, the smallest given quantity of a fluid may be made to 
Produce a pressure capable of sustaining any proposed weight, either 
oy diminishing the diameter of the column and increasing its height, 
Or hy increasing the surface which supports the weight. 

6- The perpendicular pressure of a fluid on any surface, whether 
>rertica], oblique, or horizontal, is equal to the weight of a column of 
^e fluid whose base is equal to the surface pressed, and height equal 
^o the distance of the center of gravity of that surface below the 
Opper horizontal surface of the fluid. 

7. Fluids of different specific gravities that do not mix, will 
^soanterbalance each other in a bent tube, when their heights above 
tihe surface of junction are inversely as their specific gravities. 

A portion of fluid will be quiescent in a bent tube, when the upper 
surface in both branches of the tube is in the same horizontal plane, 
or is equidistant from the earth's center. And water poured down 
one branch of such a tube (whether it be of uniform bore throughout, 
or not) will Ti»e to its own level in the other branch. 

Thus, water may be conveyed by pipes from a spring on the side 
of a hill, to a reservoir of equal height on another hill. 

8. The ascent of a body in a fluid of greater specific grarity than 
itself, arises from the pressure of the fluid upwards against the under 
surface of the body 

9. The center of pressure is that point of a surface against which 
any fluid prestes, to which if a force equal to the whole pressure 



250 PRESSURE OF FLUIDS. | 

were applied it would keep the surface at rest, or balance 
ency to turn or move in any direction. 

10. If a plane surface which is pressed by a fluid be pn 
the horizontal surface of it, and their common intersection be 
as the axis of suspension, the centers of percussion and of 
will be at the same distance /rom the axis, 

11. The center of pressure of a parallelosram, whose upp 
in the plane of the horizontal level of the liquid, is at § oj 
(measuring downwards) that joins the middles of the two } 
sides of the parallelogram. 

12. If the base of a triangular plane coincides with tl 
surface of the water, then the center of pressure is at the i 
the line drawn /rem the middle of the base to the vertt 
triangle. But, if the vertex of the triangle be in the uppe 
of the water, while its base is horizontal, Ae center of pt 
at ^ of the line drawn from the vertex to bisect the base. 

Id. If in any closed vessel containing a fluid suppose 
without weight, an opening or orifice be made and any pn 
applied, that pressure will be equally distributed over tl 
interior surface of the vessel; and if the fluid has a wei[ 
own, the pressure upon any point will equal the sum of tl 
buted pressure and the pressure occasioned by the weight of 
at that point. 

ILLUSTRATIONS AND APPLICATIONS. 



t. If several glass tubes of different shapes and 
sizes be put into a larger glass vessel containing 
water, the tubes being all open at top ; then the 
water will be seen to rise to the same height in 
each of them, as is marked by the upper surface 
a c, of the liquid in the larger vessel. 



2. If three vessels of equal bases, one cylindrical, the sec 
siderably larger at top than at bottom, the third considerab 
top than at bottom, and with the sides of the two latt< 
regularly or irregularly sloped, have their bottoms moveable, 
close by the action of a weight upon a lever ; then it will I 
that when the same weight acts at the same distance upon t 
water must be poured in to the same height in each vessel I 
pressure will force open the bottom. 

3. Let a glass tube open at both ends (whether cylindrio 
does not signify) have a piece of bladder tied loosely over < 
so as to be capable of hanging below that end, or of rising u 
it, when pressed from the outside. Pour into this tube son 
tinged red, so as to stand at the depth of seven or eight inci 



CHAP. IV.] bramah's PBESS. 251 

then imroerae the tuhe with its coloured water vertically into a larger 
glass vessel nearly foil of colourless water, the hladder heing down-^ 
wards, serving as a fiexihle bottom to the tube. Then, it will be 
observed that when the depth of the water in the tube exceeds that 
in the larger vessel, the bladder will be forced behw the tube, by the 
excess of the interior over the exterior pressure : but when the ex- 
terior water is deeper than the interior, the bladder will be thrust up 
within the tube, by the excess of exterior pressure : and when the water 
in the tube and that in the larger vessel have their upper surfaces in 
the same horizontal plane, then the bladder will adjust itself into a 
fiat position, just at the bottom of the tube. The success of this 
experiment does not depend upon the actual depth of the water in 
the tube, but upon the relation between the depths of that and the 
exterior water; and proves that in all cases the deeper water has 
the greater pressure at its bottom, tending equally upward or 
downward. 

4. The hydrostaticcU paradaof^ as it is usually denominated, results 
Prom the principle that any quantity of a non-elastic fluid, however 
ma]}, may be made to balance another quantity; or any weight, as 
arge as we please (§ 5, page 249). It may be illustrated by a 
naohine, the hydrostatic heUow9^ which consists 
»f two thick boards DC, F£, each about 16 
r 18 inches diameter, more or less, covered 
r connected firmly with leather round the 
dges, to open and shut like a common bel- 
yy/v9y bot without valves; only a pipe A B, 
boat 8 feet high, is fixed into the bellows 
boTO F. Now, let water be poured into 
lo pipe at A, and it will run into the bellows, 
radoally separating the boards by raising the 
pper one. Then, if several weights, as three hundred-weights, be 
lid upon the npper board, and water be poured in at the pipe till it 
» fnJl, it will sustain all the weights, though the water in the pipe 
bonld sot weigh a quarter of a pound : for the pipe or tube may be 
a sniall as we please, provided it be but long enough, the whole 
flTect depending upon the height, and not at all on the width of the 
ipc, for the proportion is always this: — 

As the area of the orifice of the pipe 

is to the area of the bellows board, 

so is the weight of water in the pipe, above D C, 

to the weight it will sustain on the board. 

5. In lieu of the bellows part of the apparatus, the leather of which 
rould be incapable of resisting any very considerable pressure, the 
ale Mr. Joseph Bramah used a very strong metal cylinder, in which 
I piston moved in a perfectly air and water tight manner, by passing 
.hroogh leather collars, and as a substitute for the high column of 
ivaler be adopted a very small forcing pump, to which auy power can 
be applied ; M>d thus the pressing column becomes indefinitely long. 



252 biumah's press. [part ii. 

although the whole apparatus is very compact and takes hut little 
room. Figure 207 is a section of one of tliese presses, in which t is 
the piston of the large cylinder, formed of a solid piece of metal 
turned truly cylindrical, and carrying the lower hoard v of the press 
upon it : u is the piston of the small forcing pump, being also a 
cylinder of solid metal moved up and down hy the handle or lever w. 
The whole lower part of the press is sometimes made to stand in a 
case XX, containing more than sufficient water, as at y, to Bll both 
the cylinders; and the suction pipe of the forcing pump u dip])ing 
into this water will be constantly supplied. Whenever, therefore, 
the handle w is moved upwards, the water will rise through the 
conical metal valve jt, opening upwards into the bottom of the pump 
u ; and when the handle is depressed that water will be forced 
through another similar valve a, opening in an opposite direction in 
the pipe of communication between the pump and the great cylinder 
ft, which will now receive the water, by which the piston rod t will 
be elevated at each stroke of the pump «. Another small conical 
valve c is applied by means of a screw to an orifice in the lower part 
of the large cylinder, the use of which is to release the pressure 
whenever it may be necessary ; for, on opening this valve, any w^at ei^ 
which was previously contained in the large cylinder ft, will run off 
into the reservoir y by the passage d^ and the* piston t will descend ^^ 
so that the same water may be used over and over again. Th( 
power of such a machine is enormously great; for, supposing th 
hand to be applied at the end of the handle tr, with a force of onl 
10 pounds, and that this handle or lever be so constructed as t 
multiply that force but 5 times, then the force nith which the pistoi 
u descends will bo equal to 50 pounds: let us next suppose that th< 
magnitude of the piston t is such, that the area of its horizontc 
section shall contain a similar area of the smaller piston u 50 times 
then 50 multiplied by 50 gives 2500 pounds, for the force wit 
which the piston i and the lower board v of the press will rise. J 
man can, however, exert ten times this force for a short time, am 
could therefore raise 25,000 pounds ; and would do more if a greai 
disproportion existed between the two pistons / and u^ and the lev 
w were made more favourable to the exertion of his strength. 

This machine not only acts as a press, but is capable of niai 
other useful applications, such as a jack for raising heavy loads, 
even buildings ; to the purpose of drawing up trees by their rooi 
or the piles used in bridge-building. 

To find the thickness of the metal in Bramah's press, to resi 

certain pressures, Mr. Barlow gives this theorem, t = — — whe J 

p =s pressure in lbs. per square inch, r = radius of the cylinder, i == 
its thickness, and c = 1 8000 Ihs. the cohesive power of a squ^^y 
inch of cast iron. 

£a;. Suppose it were required to determine the thickness of m^tsT 
in two presses, each of 6 inches radius, in one of which the pressure 





I 



CHAP. IV.] FRBSSURS AGAINST SLUICE-GATES, ETC. 253 

may extend to 4278 pounds, in the other to 855G pounds per square 
inch. 

Here in the first case, 

4278 X 6 .««,., t . , 

' = ,»^^^ 7x=^ = 1*87 inches, thickness. 

18000 — 4278 

In the second, 

8556 X 6 . .o . L 11 

^ = .^^^^ TTz-zz^ = 5*43 inches, thickness. 

18000 — 8556 ' 

The usual rules, explained helow (Art. 10), would make the latter 
thickness douhle the former : extensive experiments are necessary to 
tell whicli method deserves the preference. 

6. If b equals the breadth, and d the depth of a rectangular gate, 
or other surface exposed to the pressure of water from top to hottom ; 
then the entire pressure is equal to the weight of a prism of water 
inrhose content is ^ hd^. Or, if ^ and d be in feet, then the whole 
pressure = 31 J ft d\ in lbs., or nearly = -y\ b d\ in cwts. 

7. If the gate be in the form of a trapezoid, widest at top, then, 
i f B and b be the breadths at the top and bottom respectively, and d 
the depth, the 

whole pressure in lbs. = 31 J {^^ (B — b) + b} d^ 

whole pressure in cwts. = -^ {^^ (B — b) -^ b}d^ nearly. 

8. The weight of a cubic foot of rain or river water, is nearly equal 
^o -,\ cwt. 

The pressure on a souare inch, at the depth of THiR-ty feet is very 
nearly THiR-/^it pounds. 

The pressure on a square foot is nearly a ton at the depth of thirty- 
six feet. [The true depth is 35-84 feet.] 

The weight of an ale gallon of rain water is nearly 10| lbs., that 
of an imperial ^Woii 10 lbs. 

The weight of a cubic foot of •ea-water is nearly ^ of a cwt. 

These are all useful approximations ; the actual weight of a cubic 
foot of distilled water is 6*25 lbs. 

Thus, the pressure of rain water upon a square inch at the depth 
of 3000 feet, is 1300 lbs. 

And tlie pressure upon a square foot at the depth of 108 feet is 
nearly three tons. 

9. In the construction of dykes or embankments, both faces or slopes 
should be planes, and the exterior and interior slopes should make 
an angle of not less than OO"*. For, if A D' (fig. 208) be the exterior 
slope, and the angle D^ A B be acute, E D' perpendicular to A B is the 
direction of the pressure upon it; and the portion D'A E will pro- 
bably be torn off. But when DA is the exterior face, making with 
ABan obtuse angle, the direction of the pressure falls within the 
base, and therefore augments its stability. 

10. The strength of a circular bason confining water requires the 
consideration of other principles. 



254 FLOATING BODIES. [PART II. 

The perpendicular pressure against the wall depends merely on the 
altitude of the fluid, without being affected by the volume. But, as 
Professor Leslie remarks, the longitudinal effort of the thrust, or its 
tendency to open the joints of the masonry, is measured by the radios 
of the circle. To resist that action in very wide basons, the range 
or course of stones along the inside of the wall must be proportion- 
ally thicker. On the other hand, if any opposing surface present 
some convexity to the pressure of water, the resulting longitudinal 
strain will be exerted in closing the joints and consolidating the 
building. Such reversed incurvation is, therefore, often adopted in 
the construction of dams. 

Upon similar principles, the thickness of pipe$ to convey water, 

must vary in proportion to — , where h is the height of the head of 

water, d the diameter of the pipe, and c the measure of the cohe«on 
of a bar of the same material as the pipe, and an inch square. 

A pipe of cast iron^ 15 inches diameter, and \ of an inch thick, 
will be strong enough for a head of 600 feet. 

A pipe of (Hik of tlie same diameter, and 2 inches thick, woold 
sustain a head of 180 feet. 

Where the cohesion is the same, t varies B&hd; or as HD : T :: : 
hd : t^m the comparison of two cases *. 

Example, What, then, must be the respective thicknesses of pipes 
of cast iron and oak, each 10 inches diameter, to carry water from sh 
head of 360 feet ? 

Here, 1st, for cast iron: 
HD (= 600 X 15) : T (= ^) :: A rf (= 860 x 10) 
860 X 10 X S 10800 s r • ,_ 
eOOT"! 5'ir4 = 36000 = ^^ ^^ *^ '°^*^- 

2ndly, for oo^: 
HD (a= 180 X 16) : T ( = 2) :: A J (= 360 x 10) 

860 X 10 X ft ^, „ «« . , 

-i8o-^ri5- = ^ = *=^^^°^^^ 



Sect. III. Floating Bodies. 

1. If any body float on a fluid, it displaces a quantity of the fluid 
equal to itself in weight. 

* To ascertain whether or not a pipe is strong enough to sustain a proposed 
pressure, it is a good custom amongst practical men to empk>y a «^l|^e«lo#, 
usually of an indi in diameter, and load it with the proposed weight, and s 
surpluM determined by practice. Then, if the proposed pressure be aj^ied 
interiorly, by a forcing pump, or in any other way, if the pipe remain sound in 
all its parts after the safety-valve has yielded, such pipe is regarded as sufficiently 
strong. 

The aetual pressures upon a pipe of any proposed diameter and head, may 
evidently be determined by a similar method. 



CHAP. IV.J 



PARKY S FLOOD-OATE. 



255 



'2. Also, the centers of gravity of the body and of the fluid di8- 
pkieed, must, when the body is at rest, be in the same yertical line. 

8. If aressel contun two fluids that will not mix (as water and 
mercurj), and a solid of some inteimediate specific gravity be im- 
mened under the surface of the lighter fluid and float on the heavier; 
the part of the solid immersed in the heavier fluid, is to the whole 
solid aB the difference between the spedflc gravities of the solid and 
the lighter fluid, is to the difference between the specific gravities of 
the two fluids. 

4. The buoyancy of casks, or the load which they will carry with- 
out sinking, may be estimated by reckoning 10 lbs. avoirdupois to the 
ale gallon, or 8| lbs. to the wine gallon. 

5. Tlie buoyancy of pontoons may be estimated at about kalf a 
kwired-wei^ for each cubic foot. 

Thns a pontoon which contained 06 cubic feet, would sustain a 
load of 48 cwt. before it would sink. This is an approximation, in 
which the difference between -^ and |, that is, ^^ of the whole 
weight, is allowed for that of the pontoon itself. 

6. The principles of buoyancy are very ingeniously applied in 
Mr. Farcy's tdf -acting flood-gate. In the case of common sluices to 
a niil-dam, when a sudden flood occurs, unless the miller gets up in 
the nip;ht to open the gate or gates, the neighbouring lands may be* 
come inundated ; and, on the contrary, unless he be present to shut 
th«n np when the flood subsides, the mill-dam may be emptied and 
the water lost which he would need the next day. To prevent either 
of these occurrences, Mr. John Farey, whose talent and ingenuity are 
well known, has proposed a ielf-acting flood-gate^ the following de- 
scription of which has been given in the Mechanics Weekly Journal. 



— j 


_. . 


=^-^IS 


w^ 








^^ 




r^^ 




_B 


gj:^::x 






Fi 




,..,,..,^„,.„„.,,...jp.-., — — 




wL-t^ 






m~r 


V - - 




m <i 


ter 








'^ 




t^ 




.^ ,^i^^V^ — ^ — ^^^— -- — ^ " — 




1 


SA / " 








IV :/ 1 ,_>^J:^-^- 




1 


4 \y ! \ V^-i^p^ — 




r 


p 




[:■ '*~^ 



256 SPECIFIC GRAVITIES. [PART II. 

A A represents a vertical section of a gate poised upon a horizon- 
tal axis passing rather above the center of pressure of the gate, so as 
to give it a tendency to shut close: a a is a lever, fi.xed perpendicu- 
lar to the gate, and connected by an iron rod with a cask, 5, which is 
floated whenever the surface of the water rises to the line B, D, which 
is assumed as the level of the wear, or mill-dam, B,C,£, F, in which 
the flood-gate is placed : by this arrangement it will be seen that when 
the water riscK above the dam, it floats the cask, opens the gate, and 
allows the water to escape until its surface subsides to the proper 
level at B, D; the cask now acts by its weight, when unsupported 
by the water, to close the gate and prevent leakage. The gate should 
be fitted into a frame of timber, H,K, which is set in the masonry of 
the dam. The upper beam H, of the frame being just level with the 
crown of the dam, so that the water runs over the top of the gate at the 
same time that it passes through it: to prevent the current disturbing 
the cask, it is connected by a small rod, e, at each end, to the upper 
beam, H, of the frame, and jointed in such a manner as to admit of 
motion in a vertical direction. 

7> By means of the same principle of buoyancy it is, that a hollow 
ball of copper attached to a metallic lever of about a foot long, is 
made to rise with the liquid in a water-tub, and thus to close the 
cock and stop the supply from the pipe, just before the time when 
the water would otherwise run over the top of the vessel. 

8. This property, again, has been successfully employed in pulling 
up old ])i]e8 in a river where the tide ebbs and flows. A barge of 
considerable dimensions is brought over a pile as the water begins to 
rise: a strong chain which has been previously fixed to the pile by a 
ring, &c. is made to gird the barge and is tlien fastened. As the tide 
rises the vessel rises too, and by means of its buoyant force draws up 
the pile with it. 

In an actual case, abarge 50 feet long, IS feet wide, 6 deep, and draw- 
ing *2 feet of water, was employed. Here, 50 x 12 x (6 — a) x ? 

= ^Q X ^^ X 16 ^ ^^^ X 7| = 1344 + 274 = 1371f cwt.= 66J 

tons nearly, the measure of the force with which the barge acted 
upon the pile. 



Sect. IV. Specific Gravities. 

1. If a body float on a fluid, the part immersed is to the whole 
body, as the specific gravity of the body to the specific gravity of the 
fluid. 

Hence, if the body be a square or a triangular prism, and it be 
laid upon the fluid, the ratio of that portion of one end which is im- 
mersed, to the whole surface of that end, will serve to determine the 
specific gravity of the body. 

2. If the same body float upon two fluids in succession, the parts 
immersed will be inversely as the specific gravities of those fluids. 



CRAP. ly.] SPECIFIC GRAVITIES. 257 

3. The weight which a body loses when wholly immersed in a 
fluid is equal to the weight of an equal bulk of the fluid. 

When we say that a body loses part of its weight in a fluid, we do 
not mean that its abtoluie weight is less than it was before, but that it 
is partly supported by the reaction of the fluid under it, so that it 
requires a less power to sustain or to balance it. 

4. A body immersed in a fluid ascends or descends with a force 
equal to the difference between its own weight and the weight of an 
eqoa] bnlk of fluid ; the resistance or viscidity of the fluid not being 
considered. 

5. To Jind ike specific aravity of a fluids or of a solid. — On one arm 
of a balance suspend a globe of lead by a fine thread, and to the other 
fasten an equal weight, which may just balance it in the open air. 
Immerse the globe into the fluid, and observe what weight balances 
it then, and consequently what weight is lost, which is proportional 
to the specific gravity as above. And thus the proportion of the 
Q)ecific gravity of one fluid to another is determined by immersing 
the globe successively in all the fluids, and observing the weights lost 
in each, which will be the proportions of the specific gravities of the 
^mds sought. 

This same operation determines also the specific gravity of the 
Solid immerged, whether it be a globe or of any other shape or bulk, 
Supposing that of the fluid known. For the specific gravity of the 
lluia is to that of the solid, as the weight lost is to the whole 
^-eight. 

Hence also may be found the specific gravity of a body that is 
lighter than the fluid, as follows : 

O. To find the specific gravity of a solid that is lighter than the fluid y 
^i^stPOieTj in which it MjEm/.— Annex to the lighter body another that 
i^ much heavier than the fluid, so that the compound mass may sink 
in tbe fluid. Weigh the heavier body and the compound mass sepa- 
«-«tel J, both in water and out of it ; then find how much each loses 
ivi urater, by subtracting its weight in water from its weight in air; 
^nd subtract the less of these remainders from the greater. 

Then, As this last remainder, 

: the weight of the light body in air, 
: : the specific gravity of the fluid, 
: the specific gravity of that body. 

7. The specific gravities of bodies of equal weight are reciprocally 
proportionar to the quantities of weight lost in tbe same fluid. And 
tience is found the ratio of the specific gravities of solids, by weighing 
in the same fluids masses of them that weigh equally in air, and 
noting the weights lost by each. 

8. Instead of a hydrostatic balance^ a hydrostatic steelyard is now 
frequently employed. It is contrived to balance exactly by making 
the shorter end wider, and with an enlargement at the extremity. 
The shorter arm is undivided, but the longer arm is divided into 
short equal divisions: thus, if that longer arm be 8 inches long, it 



258 CAPILLARY ATTRACTION. [PART II. 

may be divided into 400 parts, the divisions wmmeneing at A (fig. 209V 
Then, in using this instrument, any convenient weight is suspended 
by a hook from a notch at the end of the scale A. The body who$e 
specific gravity is to be determined, is suspended from the other am^^ 
by a horse-hair, and moved to and fro till an equilibrium is produced^ _ , 
Then, without altering its situation at D on the beam, it is immenev:^ 
in water, and balanced a second time by sliding the couDterpois^^^ 
from A, say to C. 

Here, evidently, the weight in water : weight in air : : B C : B Am^ ^ 
and the loss of weight in water : weight in air :: AG : AB. 

^ , weight in air AB .^ 

Consequently, — — = — -- = specific gravity. 

Joss A C 

With such an instrument nicely balanced upon a convenient ped^^^ 
tal, I find that the specific gravities of solids are ascertainable b^^n^^ 
with greater facility and correctness than with any hydrostaH^t^^ 
balance which I have seen *. 

A copious table of the specific gravities of various substances "99111 
be found in the Appendix, Table X. 



Sect. V. On Capillary Attraction, 

1 . If two plates of glass be set up vertically in a shallow vessel, 
containing a coloured liquid, in such a way as to touch along the 
edges AB, fig, 210^ and forming a very acute anele vidth each other, 
the coloured liquid will rise between the two plates to a height in- 
versely as the distance between them ; from which it follows, that its 
surface will form a rectangular hyperbola, whose two asymptotes are 
the edge of the plate A B and the surface of the li(^uid. And if a 
glass tube, the bore of which is small, be immersed in the liquid, it 
will rise within the tube to a greater height than the surfiEMse of the 
liquid without the same. 

2. The cause of this phenomenon is an attractive force existing 
between the liquid and the glass, and which is termcfd capillary at- 
traction, from capiUuSy the Latin for hair, because it only Utkes place 
in tubes, the bores of which are small, resembling hairs. 

8. The heights to which fluids rise in capillary tubes are inversely 
as their diameters : and they are found to nse to the same height in 
a vacuum, as in the air. 

4. In a tube -r^n^^ ^^ ^^ ^^^^ ^° diameter, water rises 5*3 inches; 
and as the height is inversely as the diameter, their product will be a 
constant quantity, therefore 5*3 X *01 = *053 = ^, which may be 
taken to represent the attractive force between water and glaaa. The 
following are the values of ^, according to different experimenters :— - 

* We owe this coDtrivance to Dr. CotUtiy of Philaddpfaia. 



CHAP. IV.] CAPILLARY ATTBACTION. 259 

Sir I. Newton -020 

Sir D. Brevv-ster -033 

M.M.Gay Lussac -046 

Mr. Atwood '053 

5. The height at which the fluid is supported hy capillary attrac- 
tion in a tube of Yar3dng diameter, is that due to its size at the sur- 
face of the liquid, without any regard to the dimensions or shape of 
the lower portion. Thus, in a vessel of the form shown in figure 211, 
terminating in a capillary tube, the liquid will be supported at the 
same heifi;ht in this tube (and also fill the whole of the space abc of 
the TesscI) as if the tube had been of uniform dimensions throughout 
its whole length, and every where equal to its upper portion. The 
water, however, contained in the lower portion of the vessel a & c, is 
in this case supported by the pressure of the atmosphere, as it ceases 
to be supported in a vacuum. 

6. Different liquids rise to different heights in capillary tubes of 
the aaxne bore, depending upon the attraction between them and 

^lass. The following are Dr. Brewster's results for several fluids, 
irith a tube '0561 inches in diameter : — 

Height of Value of the 

aaoent in inchef. constant q. 

Water -587 0327 

Water, very hot 537 0301 

Muriatic acid 442 0248 

Nitric acid -395 -0222 

Spermaceti oil -392 0220 

Olive oil -387 '0215 

Oil of turpentine -333 -0187 

Alcohol -317 -0178 

iEther -285 -0160 

Sulphuric acid 200 0112 

7. The internal diameter of a uniform capillary, or other small 
tube, may be found in the following manner. Let the tube be weighed 
when empty, and again when filled with mercury, and let w be the 
difference of those weights in troy grains, / the length of the tube in 

inches; and d its diameter, then d = '019252 \/-j 

Thos, if the difference of the weights were 500 grains, and the 
length of the tube were 20 inches: we should have d = *010252 

y^^ = -019252 X 5 = -09626 of an inch* 

* Tlie same thing may easily i>e aeoomplithed thot t — Let a cone of box wood, 
0r of bran, be very aoeunitely turned, or about 6 inches in length, and the dia- 
naecer of its base aboat a quarter of an inch ; and let its curve surface i>e very 
aecoratelj marked with a series of parallel rings, about a twentieth of an inch 
aa i mde r, from its vertex to its base. Insert this cone carefully in the cylinder 
(ao that their axes shall coincide) as in fig. 212 : then it will be as V A i Va : t 
A B : eft; where, as the ratio of V A to V a is known by means of the equi- 
distant rings on the iorfaea, and A B is known, a b becomes determined. 

82 



260 SPFLUBNC£ OF FLUIDS. [pABT II. 



CHAP. V. 
HYDRODYNAMICS. 

HydrodynamicB is that part of mechanical science which relates to 
the motion of non-elastic fluids, and the forces with which they act 
upon bodies. 

This branch of mechanics is the most diflicult, and the least ad- 
vanced : whatever we know of it is almost entirely due to the re- 
searches of the modems. 

Could we know with certainty the mass, the figure, and the num- 
ber of particles of a fluid in motion, the laws of its motion might be 
determined by the resolution of this problem, viz. to find the motion 
of a proposed system of small free bodies acting one upon the other 
in obedience to some given exterior force. We are, however, ▼cry 
far from being in possession of the data requisite for the solution of 
this problem. We shall, therefore, simply present a few of the most 
usually received theoretical deductions ; and then proceed to state 
those rules which have resulted from a judicious application of theory 
to experiment. 



Skct. I. Motion and Effluence of Liquids. 

1 . A jet of water, issuing from an orifice of a proper fonn, and 
directed upwards, rises, under favourable circumstances, nearly to the 
height of the head of water in the reservoir; and since the particles 
of such a stream are but little influenced by the neighbouring ones, 
they may be considered as independent bodies, moving initially with 
the velocity which would be acquired in falling from the height of the 
reservoir. And the velocity of the jet will be the same whatever 
may be its direction. 

2. Hence, if a jet issue horizontally from any part of the side of 
a vessel standing on a horizontal plane, and a circle be described 
having the whole height of the fluid for its diameter, the fluid wiU 
reach the plane at a distance from the vessel, equal to that chord of 
the circle m which the jet initially moves. 

Thus, if AS (fig, 213) be the upper surface of the fluid in the ves- 
sel, B the place of the orifice, CF the horizontal plane on which the 
fluid spouts, then CF is equal to £D, the horizontal chord of the 



BPFLUXNCE OF FLUIDS. 261 

le diameter is A C, passing throagh B. It is therefore evi- 
tbe orifice from which the fluid will spout to the greatest 
I situate at G, half the height of the fluid, and also, that if 
ires be made at equal distances ahoTC and below 6, the jet 
om both will strike the plane C £ in the same point. 
m a cylindrical or prismatic vessel empties itself by a small 
3 Telocity at the surface is uniformly retarded; and in the 
nptying itself, twice the quantity would be discharged if it 
I full by a new supply. 

the quantity dischsrged is by no means equal to what would 
lole orifice, with this velocity. If the aperture is made sim- 
lin plate, the lateral motion of the particles towards it tends 
i the direct motion, and to contract the stream which has 
ifice, nearly in the ratio of two to three. So that, in order 
le quantity discharged, the section of the orifice must be 
to be diminished from 1 00 to 62 for a simple aperture, to 
ipe of which the length is twice the diameter, and in other 
»rding to circumstances. 

m. a syphon, or bent tube, is filled with a fluid, and its 
mersed in the fluids of different vessels, if both surfaces of 
are in the same level, the whole remains at rest ; but if 
the longer column of fluid in the syphon preponderates, 
■easure of the atmosphere forces up the fluid from the higher 
til the equilibrium is restored ; and the motion is the more 
be difference of the levels is greater: provided that the 
^ht of the tube above the upper surface be not more than 
loise to the pressure of the atmosphere. 
le lower vessel be allowed to empty itself, the syphon will 
mining as long as it is supplied from the upper, and the 
it descends the further below the vessel. In the same 
e discharge of a pipe, descending from the side or bottom 
i vessel, would be increased almost without limit by length- 
notch or sluice in form of a rectangle be cut in the ver- 
of a vessel full of water, or any other fluid, the quantity 
loagb it will be ^ of the quantity which would flow through 
nifice, placed horizontally at the whole depth, in the same 
reasel being kept constantly full. 



ivr«Deiit in the oonstniction of the syphon has been lately proposed 
pnv Mechanics^ Magazine^ and by M. burUem at Paris. It might 
mtageouftly used if constructed on a larse scale, for lowering the 
B-4laiBS or canals. The improvement in toe present syphon is, that 
pipe is enlarged to the same diameter as that of the syphon, and 
lened out to something of a funnel shape, as in fig. 214. In 

J action, the short arm is immersed in the water as in the usual 

bottom of the long arm is closed, the exhausting pipe is then filled 
hjr the funnel-^aped mouth. On the bottom of the long arm being 
Mtter flows out, and exhausts the air from the syphon, when the 
rh wished to be emptied flows out in a continual stream. 



z& 



i 



262 PIPB8 AND CANALS. [PART II. 

8. If a short pipe elevated in any direction from an aperture in t 
condnit, throw the water in a parabolic curre to the distance or range 
R, on a board, or other horizontal plane passing through the orifice^ 
and the greatest height of the spouting fluid above that plane, be H, 
then the height of the head of water above that conduit pipe, may be 

found nearfy: vis. by taking 1st, 2 cot £ = ^--; and 2ndly, the alti- 

2 n 

tude of the head A = i R x cosec 2 E. 

Example. Suppose that R = 40 feet, and H = 18 feet. Then 

— == ~ = Mllllll =2cot60*'57': and A = i R x cosec2E 

= 20 X cosec 121* 5V = 20 x 1-177896 = 23-55792 feet, height 
required. 

Note. This result of theory will usually be found about ^ of that 
which is furnished by experiment 



Sbci. II. Motion of Water in Conduit Pipes and Open Canals^ 
over Weira^ 4^. 

1. When the water from a reservoir is conveyed in long horizontal 
pipes of the same aperture, the discharges made in equal times are 
nearly in the inverse ratio of the square roots of the lengths. 

It is supposed that the lengths of the pipes to which this rule is 
applied, are not very unequal. It is an approximation not deduced 
from principle, but derived immediately from experiment. [Bossut, 
tom. ] 1, $ 647, 648. At § 673, he has given a table of the actual 
discharges of water-pipes, as far as the length of 2340 toises, or 
14,950 feet English.] 

2. Water running in open canals, or in rivers, is accelerated iu 
consequence of its depth and of the declivity on which it runs, till 
the resistance, increasing with the velocity, becomes equal to the 
acceleration, when the motion of the stream becomes uniform. 

It is evident that the amount of the resisting forces can hardly be 
determined by principles already known, and therefore nothing 
remains but to ascertain, by experiment, the velocity corresponding 
to different declivities, and different depths of water, and to try, by 
multiplying and extending these experiments, to find out the law 
which is common to them all. 

The Chevalier Du Buat has been successful in this research, and 
has given a formula for computing the velocity of running water, 
whether in close pipes, open canals, or rivers, which, though it may 
be called empirum^ is extremely useful in practice. Principea 
d'Hydrauligue. Professor Robison has given an abridged account of 
this book, m his excellent article on Rivers and Water-worics, in the 
JEnejfdopcadia Britannica. 

Let V be the velocity of the stream, measured by the feet it moves 
over in a second ; R a constant quantity, vis., the quotient obtained 



CHAP, v.] PIPES AND CANALS, 263 

by dividiog the area of the transverse section of the stream, expressed 
in square feet, by the boundary or perimeter of that section, minus 
the saperficial breadth of the stream expressed in linear feet. 

The mean velocity is that with which, if all the particles were to 
move, the discharge would be the same with the actual discharge. 

The line R is <^Ied by Dn Buat, the raditi$; and by Dr. Robison, 
the k^raulie mecm depth. As its affinity to the radius of a circle 
eeenifl greater than to the depth of a river, we shall call it, witli the 
former, the radiui qfihe section, 

LasUy, let 8 be the denominator of a fraction which expresses the 
slope, the numerator being unity, that is, let it be the quotient ob- 
tained by dividing the length of the stream, supposing it extended in 
^ straight line, by the difference of level of its two extremities : or, 
-virhich is nearly the same, let it be the co-tangent of the inclination 
or slope. 

3. The above denominations being understood, and the section, as 
-well as the velocity, being supposed uniform, we have 

When R and 8 are very great, 

^ = »'{sn^.-4' ""''^^ ^"-^ 

The logarithms understood here are the hyperbolic, and are found 
by multiplying the common logarithms by 2*3025851 ; or more easily 
by the method described at page 94. 

The slope remaining the same, the velocities are as n/r — y^. 

The velocities of two great rivers that have the same declivity, are 
as the square roots of the radii of their sections. 

If R is so small, that n/R — -^j^ = 0, or R = -^j, the velocity 
will be nothing; which is agreeable to experience; for in a cylindric 
tube R =s ^ Uie radius ; the radius, therefore, equals two-tenths ; 
so that the tube is nearly capillary, and the fluid will not flow 
through it. 

The vdodty may also become nothing by the declivity becoming 
so small, that 

if - is less than — — —-, or than Vn^h of an inch to an English 
8 600000' ;^ ^ 

mile, the water will have sensible motion. 

4. Ib a river, the greatest velocity is at the surface, and in the 

middle of the stream, from which it diminishes towards the bottom 



2S4 PIPES AND CANALS. [PABT II. 

and the sides, where it is least. It has been found by experiments 
that if from the square root of the Telocity in the middle of the 
stream, expressed in inches per second, unity be subtracted, the 
square of the remainder is the velocity at the bottom. 

Hence, if the former velocity be = 9, the velocity at the bottom 
c=r— 2v^t?4-l (HI.) 

5. The mean velocity, or that with which, were the whole stream 
to move, the discharge would be the same with the real discharge, is 
equal to half the sum of the greatest and least velocities, as computed 
in the last proposition. 

The mean velocity is, therefore, =»— ^/c-f^ (IV.) 

This is also proved by the experiments of Dn Buat. 

6. Suppose that a river having a rectangular bed, is increased by 
the junction of another river equal to itself, the declivity remaining 
the same ; required the increase of the depth and velocity. 

Let the breadth of the river = 5, the depth before the junction </, 
and after it d?; and in like manner, v and v^ the mean velocities 

before and after; then ^ , is the radius before, and 

b + 2d ' 6 + 2x 

the radius after, so o = r — , supposing the breadth of the river 

to be such, that we may reject the small quantity subtracted from R, 

SOT R * 
in formulce (I. and II.); and, in like manner, v^ = r— !-. 

Then, substituting for R and R^, we have 

^^30T , f~hd 






S* ^ ^ cH- 2<^' 

307 / vof 
and c. =--T- X A/ -— . 

Multiplying these into the areas of the sections 5 </ and ior, we 
have the discharges, viz., 

307 hds/hd ^^ 307 hxs/hx 

Now the last of these is double of the former ; therefore, 
hx^hx 2hds/hd x" 4rf» 




>/^ -f 2rf' 'ft-h2« 64-2rf' 



and 



t TT-k ^ = 1 r-i J a cubic equation which can always 

be resolved by Cardan's rule, or by the approximating method given 
at page 82. 

As an example, let ft = 10 feet, and </ = 1, then x^ — J « = ^*>, 
and X = 1*4882, which is the depth of the increased river. Hence 
we have 1-488 x t>j = 2 r, and 1'488 : 2 : : «? : Cj, or t? : ©j : : 37 
to 50 nearly. 



CHAP, v.] 



CANALS, RIVKBS, &C. 



265 



When the water in a river receives a permanent increase, the 
depth and the velocity, as in the example above, are the first things 
that are augmented. The increase of the velocity increases the 
action on the sides and bottom, in consequence of which the width is 
aaemented, and sometimes also, but more rarely, the depth. The 
velocity is thus diminished, till the tenacity of the soil or the hard- 
ness of the rock affords a sufficient resistance to the force of the 
^water. The bed of the river then changes only by insensible 
degrees, and, in the ordinary language of hydraulics, is said to be 

permanent, though in strictness this epithet is not applicable to the 

course of any river. 

7' When the sections of a river vary, the quantity of water 

s-emaining the same, the mean velocities are inversely as the areas of 

^he sections. This must happen, in order to preserve the same 

quantity of discharge. (Pla^air's Outlines,) 

8. The following table, abridged from Du Buat, serves at once to 

compare the surface, bottom, and mean velocities in rivers, according 

t:o the formul® (III. and IV.) 



VdocUies of Rivers. 



VRI/K31TY IN INCHES. 


VELOCITY IN INCHES. 


Sm&ce. 


Bottom. 


Mean. 


Snr&ce. 


Bottom. 


Mean. 


4 


1- 


2-5 


56 


42-016 


49-008 


8 


3-342 


5-67 


60 


45-509 


52-754 


la 


6071 


9-036 


64 


49- 


56-5 


16 


9- 


12-5 


68 


52-505 


60-252 


20 


12065 


16-027 


72 


56-025 


64-012 


24 


15194 


19-597 


76 


59-568 


67-784 


28 


18-421 


23-210 


80 


63-107 


71-553 


32 


21-678 


26-839 


84 


66-651 


75-325 


36 


25- 


30-5 


88 


70-224 


79112 


40 


28-345 


34-172 


92 


73-788 


82-894 


44 


31-742 


37-871 


96 


77-370 


86-685 


48 


35151 


41-570 


100 


81- 


90-5 


52 


38-564 


45-282 









9. The Icnowledge of the velocity at the bottom is of the greatest 
use for enabling us to judge of the action of the stream on its bed. 

Every kind of soil has a certain velocity consistent with the sta- 
bility of the channel. A greater velocity would enable the waters to 
tear it up, and a smaller velocity would permit the deposition of more 
movable materials from above. It is not enough, then, for the sta- 
bility of a river, that the accelerating forces are so adjusted to the 



260 CANALS, RIVBBSy &C. [PART II. 

size and figure of its channel that the current may be in train : it most 
also be in equilibrio with the tenacity of the channel. 

We learn from the obseryations of Du Buat, and others, that a 
velocity of three inches per second at the bottom will jast begin to 
work upon the fine clay fit for pottery, and howerer firm and com- 
pact it may be, it will tear it up. Yet no beds are more stable 
than clay when the velocities do not exceed this: for the water 
soon takes away the impalpable particles of the superficial clay, 
leaving the particles of sand sticking by their lower half in the 
rest of the clay, which they now protect, making a very perma- 
nent bottom, if the stream does not bring down gravel or coarse 
sand, which will rub off this very thin crust, and allow another layer 
to be worn off; a velocity of six inches will lift fine sand; eight 
inches will lift sand as coarse as linseed ; twelve inches will sweep 
along fine gravel ; twenty-four inches will roll along rounded pebbles 
an inch diameter; and it requires three feet per second at the bottom 
to sweep along shivery angular stones of the size of an egg. (i2o6i- 
son on Rivers,) 

1 0. In the elbow or bend of a river, the velocity is always greater 
near the concave than the convex side. 

1 1 . The swell occasioned by the piers of a bndge, or the sides of a 
cleaning sluice which contract the passage by a given quantity, for a 
short length only of the channel, may be determined when the ver- 
tical section of the river and the velocity of the stream are known, in 
the following manner. Let v be the velocity of the stream, inde- 
pendently of the effect of the bridge, r the section of the river, and a 
the amount of the sections between the piers; let 2^, instead of 
being taken at 64j, be reckoned 58'6, to accord with the results of 
experimental contractions through arches of bridges, &c., and let $ 
be the slope of the bed of the river, or the sine of its angle with the 
horizon ; then Du Btiat (tom. i. p. 225) gives for the swell or rise 
(R) of the stream in feet, which will be occasioned by the ob* 
struction. 

-(3^0-) (GI-); •• ■<^' 

12. The value of s will, of course, be different in different cases; 
but if we assume -^ or '05, as a mean value, it will enable us to 
compute and tabulate results, which, though they cannot be pre- 
sented as perfectly correct, may be regarded as exhibiting a medium 
between those that will usually occur; and will serve to anticipate 
the consequences of floods of certain velocities, when constrained to 
pass through bridges which more or less contract the stream. 



CHAP, v.] 



SWELLS 0C0A8I0NED BY BRIDGES. 



267 



TaUe of the Rise of Water occaeumed hy Piere of Bridges^ or 
other Contractions^. 





Amoum of otHtTiictiaafi f^inupved with th* vortical HetJon of the Hiver. 




MOth. 


a^iothi. 


3-lOlhA 


40)>thi. a-ioth». 


B-ltrthj. 


7'imhi. 


a-ieth.. 


o^iotiii. 




4 


Pruporliotml Riie of W»tcr, in f«rt and decimals. 






feel. 


tbe^ 


rwt 


fwt. 


f«t. 


fc«t. 


fMt. 


feeu 


fMt. 




<K11S7 


(M)377 


^HMlMr 


0*[IS2 


OS<H!t 


0-3631 


(1^78 


l^gw 


mam 






*-os77 


<]^)IW5 


flisai 


OS 102 


«':t'^#4« 


ivn^^ 1 


1199S 


ia37a 


\i1\m 


\OTtUll«T7 

J flex>di. 




^NHT? 


IM|*4 


0^1 in 


D-aiSlfl 


«^JfP7 


IHlfiHT 


MSiM> 


4i»MI 


20-1 A>4 




(H*70l 


i>18» 


frasTf 


mi^ 


{*^iti 


|-7iM«i 


3't7&5 


r-Tjan 


3tm^> 




ft-UfiS 


(^'^709 


i^sim 


fl'STUS 


i*4aafl 


2-6<ifpfi 


5i>if>2 


il'ftlff- 


4ft- 1555 


> Violent 
1 flfxirti. 




criAsa 




0^19 


v\m 


iijyi?5 


:!-Mifiti 


d7i^ 


iMtlH 


tts-7sie 




tP-i«7« 


C^4<m: 


(H»iSl 


i-57a> 


sri579 


4-6511 


a-SL^iTe 


n-mm 


Ji7'TiW> 




ri-iii78 


O-eiia 


11BB4 


iteso 


3*25* 


aawT 


H^AIM 


^Mm 


li3tM±i 


I t'lJiit'iiPilly 




033iO 


^"'N>54 


IMi^JO 


2-55Ge 


A'W^ 


7'MT3 


14^4777 


34a!J4e 


Ul-JSAi 


J. vjolcnt 


10 


0^41 19 


\mn 


i-mj^ 


^'li^ie 


s^asw 


9^tJt» 


J77M1 


4a*i+MJ 


iTk^um 


1 floodi. , 



13. It will be eyident, from an inspection of this table, that even 
in the case of ordinary floods, old bridges with piers and starlings, 
occupying 6 or 7-tentbs of the section of the river, will produce a 
swell of 2, 3, or more feet, often overflowing the river^s banks and 
occasioning moch mischief. Also, that in violent floods, an obstmc- 
tion amounting to T-tenths of the channel, will cause a rise of 7 or 8 
feet, probably choking up the arches and occasioning the destruction 
of the bridge. Greater velocities and greater contractions produce a 
rapid augmentation of danger and mbchief ; as the table obviously 
shows. 

14. The same principles and tabulated results «erve to estimate 
the fall from the higher to the lower side of a bridge, on account of 
an ebbing tide, &c. Thus, for old London Bridge, where the 
breadth of the Thames is 926 feet, and the sum of the water ways 
at low water only 236 feet, the amount of the obstructions was 690 
feet, about 7| tenths of the entire section : so that a velocity of 
3 1 feet per second would give a fall of nearly 4| feet, agreeing with 
the actual result. 

At Rochester Bridge, before the opening of the middle arches, the 
piers and starlings presented an obstruction of 7-tenths, and at the 
time of greatest fall> the velocity 100 yards above bridge exceeded 

6 feet per second. This, from the table, would occasion a fall of 
more than 6-7 feet ; and the recorded results vary from 6| to 

7 feet 

At Westminster Bridge, where the obstructiona are about l-sixth 
of the whole channel, when the velocity is 2^ feet, the fall but 
little exceeds half an inch: a result which the table would lead 
us to expect. 

•. A timilar table was computed by Mr. Wright of Durham, more than 
ftfty jmn ago, and inierted in the first edition of Dr. Hutton*B treatiM on 
Bndget ; but it it not constructed upon a correct theory. 



268 eytblwbin's bbsults. [part u. 

15. Mr. Eytelwein^ a German mathematiciany has devoted much 
time to inquiries in hydrod3mamic8. In his investigations he has 
paid attention to the mutual cohesion of the liquid moleculee, their 
adherence to the sides of the vessel in which the water moves, and 
to the contraction experienced by the liquid vein when it issues from 
the vessel under certain circumstances. He obtains formulse of the 
utmost generality, and then applies them to the motion of water; 
1st, in a cylindric tube; 2ndly, in an open canal. 

16. Let d be the diameter of the cylindric tube EF (fig. 215), 
h the total heieht F G of the head of water in the reservoir above the 
orifice F, and ? the length £ F of the tube, all in inches : and v equal 
the velocity in inches per second with which the fluid will issue 
from the orifice F ; then 



V • 



^'iV'-f^ <"•> 



which, multiplied into the area of the orifice, will give the quantity 
discharged per second. 

1 7. Let d = the diameter of the pipe in inches, Q = the quantity 
of water in cubic feet discharged through the pipe per minute, / = 
the length of the pipe in feet, and h = the difference of level between 
the surface of the water in the reservoir and at the end of the pipe, 
or the head; then, any three of these quantities being given, the 
fourth may be determined from the follo\idng formulae: — 

^= ^'O-^^^Q-C^^^^-^^ (VII.) 



« == V^tt^^^^^t:^ (VIII.) 



hd' 

•0448 (/ + 4-2 d) 



I = ^^' , - 4 2rf (IX.) 

•0448 Q'* ^ ^ 

. -0448 Q2(/ + 4-2 rf) .„^ 

^= ^^5 (^-^ 

These formulae are more convenient when expressed logarithmic- 
ally, and then become 

log (/ = J {2logQ +2-6515 + log(/ + 4-2(/) -logA} (XL) 

logQ = I {logh + 5\ogd-^ 2-6515 - log(/ + 4-2flO} ... (XIL) 
log / =log;i+5logrf-2'-6515-2logQ,[neglecting — 4-2flri(XIII.) 

log A = 2 logQ+ ■2-6515 + log (/ + 4-2 (/) - 5 log rf (XIV.) 

18. When a pipe is bent in one or more places, then if the squares 
of the sines of the several changes of direction be added into one 
sum «, the velocity v will, according to Lang^dorf^ be foond by 
the theorem 



CHAP. 



v.] 



bttblwbin's bbsults. 



269 



= a/^ 



SMdk 



(XV.) 



■^l.+ ids 

/, k^ d^ and 9, being all in inches. 

19. For open canals. — Let v be the mean velocity of the current 
in feet, a area of the vertical section of the stream, p perimeter of 
the section, or snm of the bottom and two sides, / length of the bed 
of the canal corresponding to the fall hy all in feet : then 

* = /^ 9582 — + 00111 - 0-109 (XVI.) 

The experiments of M. Bidone^ of Turin, on the motion of water 
in canals, agree within the 80tli part of the results of computations 
from the preceding formulae. 

20. For apertures in the sides or bottom of vessels,— \^ q equals the 
€]iiantity of water discharged in cubic feet per minute, v the velocity 
of the effluent water in feet per second, through the aperture, a the 
sreA of the aperture in square inches, and h the height from its center 
to the surface of the water ; we have 

v^Cs/h (XVII.) 

y=-4ie7ac ^h (XVIII.) 

In which c is a constant quantity depending upon the nature of the 
aperture, and the value of which for several different forms is con- 
tained in the following table. 



Nature of the Orifices employed. 


Ratio between 

the theoretical 

and real 


Coefficients 

for finding 

the velocities 

in Eng. feet. 


For the whole velocity due to the height 

For an orifice of the form of the contracted > 
vein. 

For wide openingi whose bottom is on a 1 
kvd with that of the reservoir ; for sluices ( 
with walls in a line with the orifice ; for T 
bridges with pointed piers. 

For narrow openings whose bottom is on a^ 
levd with that of the reservoir; for 
mailer openings in a sluice with side - 
waDa ; for abrupt projections and square 
piers of bridges. J 

For openinp in sluices without side walls 

For sui orinoe in a thin nlate 


1 to 100 
1 to 0-969 

1 to 0-961 

1 to 0-861 

1 to 0-635 
1 to 0-621 

1 to 0-510 

1 to 0-671 
1 to 0-808 


80458 
7 8 

7-7 

69 

51 
5-0 

41 

5-4 
6-5 


When there is a short cylindrical pipe pro- ) 
jeeting on the inside of the vessel, length ( 
9 to 4 times the bore, and rim like con- f 
tracted vein, not fulL J 

Idem, when it runs fuU bore of water 


Idem, when it does not project inside the \ 

VCMCL S 



MOTION IN PIPI89 BTC. 



[PA&l 






r 









t 

-I 



Bore 
of the 
pipe in 
inches. 


'*"*'p^ :I?:?2'« oSTSToo « ^ ?.o»i^»«©-2 




.SO 

1- 


S 2 2 ZS ^ * *^* *® "* '* •* ** ** ** ** '^ '^ ^ '^ '^ 




6 « © w lb r»b « ©• Oft » 00 r»^ « '^ ^ 


1- 


© © ep -* «o © 00 ^© t^oo "* 00 cc* « »^r*9 
t^^oot^««'<«eoeoc«G«e4p^i-«^-4^©©©c 


a. 

1 

a 

1 


1" 


© rt ©© *-©©-*(» rt ^ -5 (N :• »^ »-« -r- <?•« « 

00 — ©to©6>o©ooeo^oo©<«^^©o»aor<»cs 
eo©©kfd^MMeoG«M*iMpi4^PH^©©©©©© 


.So 

1- 


©©ie'«eo©oi©e4e4eo-4<«-4^©or<»©ie© 
•^do©©^©« &i©69ib99<^©©0Dt^<b^«ib 

©©«^oooi&«e«ei-«'^'^-H^©©©©©©© 


•0 

S 
•3 

> 


a CO 

fa 

.go 

fa 


t>.N©eO©iO'*0 — C0M^»O» — — '*00«9»« 

•^©o© ^©©r^ib&«©©©Qor«©>e<«'^ei»f 
t^-^eowe^N-^ — »- — — ©o©©oo©©© 

»^»o©P-©« o"** 10© M ©!'*«« — ©«»«r 
©N-iy — r^'*e«©f^©oor*o «©'*'* eo«*ir 

^0094^-4 — -«^©©©o©©©©©©© 


d«o 

1- 


© CO 10 © t^O »^0D © © •?• © «p © •© W © t 

*:- ^ lb 91 ^ © 00 r«»^ •« kid ^ CO eo 99 e« o« -N .. 

OI^pHp-©©©©©©©©©©©©00 


.So 

1- 


»^— '*t>.©«eit-cp»'?''r'»^T*'?'9f 

;ii,:^QO©ib'ii"^ooeoN©i«'^'-^-*'^-^< 

^o©©©oo©©©©oo©ooe 


.S«o 


10 © <N 00 10 CO - © © r^o « '^ "^ « o' 
;,«©,^^;^^-©©©©©©©©< 

©©©©©©oo©©o©©©e© 


Bore of 
the pipe 

'in 
inches. 


'*"*'^ :ir;r2'N srs?co So ^ ^-5 « t 



CHAP, v.] 



TABLBS, &C., FOB WEIRS. 



271 



Look for the Telocity of water in the pipe in the upper rew, and 
io the colamn below it, and opposite to the given diameter of the 
pipe studing in the last column, will be found the perpendicular 
ha^ of a column or head, in feet, inches, and tenths, requisite to 
orercome the friction of such pipe for 100 feet in length, and obtain 
the giren Telocity. 

22, TaUe containing the qttantity of Water discharged over an inch 
vertical section of a Weir. 



Depth of the 
upper edge 
of the waste- 
boaid below 
theiur&ce 
in English 
inches. 


Cnbic feet of 
water discharged 
in a minute by an 
inch of the waste- 
board, according 

to Du Buat's 
formuls. 


Cubic feet oi 
water discharged 
in a minute by an 
inch of the waste- 
board, according 
to experiments 
made in Scotland. 


Gallons 

of 

282 inches, 

corresponding 

with 

results in 

col. 3. 


1 


0-403 


0-428 


2-621 


2 


1-UO 


1-211 


7-417 


3 


2-095 


2-226 


13-634 


4 


8-226 


3-427 


20-990 


5 


4-507 


4-789 


20-332 


6 


5-925 


6-295 


38-357 


7 


7-466 


7-983 


48-589 


8 


9-122 


9-692 


59-364 


9 


10-884 


11-664 


70-826 


10 


12-748 


13-535 


83-164 


11 


14-707 


15-632 


95-746 


12 


16-758 


17-805 


109-055 


13 


18-895 


20-076 


122-965 


14 


21-117 


22-437 


137-427 


15 


23-419 


24-883 


152-408 


16 


25-800 


27-413 


167-906 


17 


28-258 


80024 


183-897 


18 


30-786 


82-710 


200-350 



^^ To the aboTe table, originally due to Du Buaty is added a third 

^^QiDD, containing the quantities of water discharged, as inferred 

J^Jffl experiments made in Scotland, and examined by Dr. Robison, 

Jo found that they in general gaTe a discharge -^ greater than that 

^lich » deduced from Du BucSs formulas. We would recommend 

^ therefore to the engineer to employ the third column in his prac- 

^^ or the fourth if he wish for the result in gallons. 

. Ifdiej be odd quarters of an inch, look in the tabic for as many 

^^<ie« as the depth contains quarters, and take the eighth part of 

^inswer. Thus, for 3| inches, take the eighth part of 24-883, 

^luch corresponds to 15 inches. This is 3*110. 

23. The quantity discharged increases more rapidly than the width : 
^ obUin a correct measure of it, if / be the width or length of tho 



272 DI8CHAROX8 OVBB WEIBS. [PABT 11. 

waste-board in inches, take (i-^-^l) times the quantity for one inch of 
wasteboard of the given depth, from the preceding table. 

In the preceding table it is supposed that the water from which the 
discharge is made is perfectly stagnant; but if it should happen to 
reach the opening with any velocity, we have only to multiply the 
area of the section by the velocity of the stream. 

24. When the quantity of water Q discharged over a weir, is 
known, the depth of the edge of the wasteboard, or H, may be 
approximated from the following formulae, / length of wasteboard. 

Q = 11-5/H^ nearly (XX.) 

or, more accurately by adding the correction in article 15, 

Q = ii-5(^ + 4) «* (^Xl) 

26. The quantities discharged for any given width are as the 

^ power of the depth, or as H *. 

Hence, to extend the use of the table to greater depths, wc have 
only for 

Twice any depth, take Q x 2*828 

3 times Q x 5196 

4 times Q x 8*000 

5 times Q x 1M80 

6 times Q x 14-697 

7 times Q x 18*520 

8 times Q x 22*627 

9 times Q x 27000 

10 times Q x 31-623 

and the results will be nearly true. 

To make them still more correct, where great accuracy is required, 
add to them their thotisandth part. 

Easamples of the use of* the Tables and Rules. 

Ex, 1. Let the depth of a weir be 10 inches below the upper 
surface of the water, and the width 8 inches. How many cubic feet 
of water will be discharged in a minute ? 

cub.fiB«t 

By table § 23. Q for 10 inches deep and 1 inch wide = 13*535 
Multiply this by / == 8 



106-280 
Add ^>jy of this product 5*314 



Discharge in one minute = 111*594 



CHAP. V.j STRBAli liSASUKSBt. 97d 

Em. S. Let the depth be 9 feet, and the width 1 fooL Required 
the cubic feet discharged in a minute. 

Bj the table Q for 12 inches deep and 1 inch wide s 17*805 

Faetor for 9 times depth ss 27 

Quantity for 1 inch width 480*735 

Multiply by / » 12 

5768-802 
Add ^9 of the product 266*441 

Total quantity in cubic feet s= 6055*261 



Ex, 3. Let a square orifice of 6 inches each side be placed in a 
sluice-gate with its top 4 feet below the upper surface of the water : 
how much will it discharge in a minute ? 

Here the quantity discharged by a slit in depth 48 inches, must be 
taken from one in depth 54 inches. 

cubic feet. 

For 54, multiply Q for 6 by 3' or 27 169*965 

For 48, Q for 12 by 4^ or 8 142440 

Difference 27*525 

S7*525 X (6 + ^) as 173*4 cubic feet, quantity discharged. 

i^(C^.<— In an example like this, it is a good approximation, to 
9nuUiyUf confinuaify together the area of the orifice^ the number 336*, 
€^n£l ike equare root of the depth infect of the middle of the orifice. 

Thus, in the preceding example, it will be | x ^ x 336 x >/4*25 
3= ^ X 336 X 2*062 = 173*2 cubic feet. 

The less the height of the orifice compared with its depth under 
^e water, the nearer will the result thus obtained approach to 
tte truth. 

If the height of the orifice be such as to require consideration, the 
2Mnciple off 7, page 261, may be blended with this rule. 

Thnsy 9km\jmg ^is rule to Ex. 2, we shall have area x s/ depth 

^ dd6 X I ss 9 X 3 X 224 = 6048, for the cubic feet discharged. 

^niis is less than the former result by about its 900th part. It is, 

ttlierefore, a good approximation, considering its simplicity: it may in 

lEKianj VMes supersede the necessity of recurrence to tables. 



Skcy. III. Coniriveaices to measure the velocity of running waters. 

1. For these purposes, various contrivances have been proposed, of 
^hieh two or three may be here described. 



836 « 6 6 x 60. 



274 STREAM MBASCREBS. [PART II. 

Suppose it be the Telocity of the water of a river that is required ; 
or, indeed, both the Telocity and the quantity which flows down it in 
a given time. ObserTe a place where the banks of the nTer are 
steep and nearly parallel, so as to make a kind of trough for the 
water to run through, and by taking the depth at Tarious places in 
crossing make a true section of the nTer. Stretch a string at right 
angles OTer it, and at a small distance another, parallel to the first 
Then take an apple, an orange, or other small ball, just so much 
lighter than water as to swim in it, or a pint or quart bottle partly 
filled with water, and throw it into the water aboTO the strings. 
ObserTe when it comes under the first string, by means of a quarter- 
second pendulum, a stop watch, or any other proper instrument; 
and obsenre likewise when it arrives at the second string. By 
this means the Telocity of the upper surface, which in practice 
may frequently be taken for that of the whole, will be obtained. 
And the section of the riTer at the second string must be ascer- 
tained by taking various depths, as before. If this section be the 
same as the former, it may be taken for the mean section: if not, 
add both together, and take half the sum for the mean section. 
Then the area of the mean section in square feet being muldplied 
by the distance between the strings in feet, will give the contents 
of the ^'ater in solid feet which passed from one string to the 
other during the time of observation : and this, by the rule of three, 
may be adapted to any other portion of time. The operation may 
often be greatly abridged by taking notice of the arrival of the float- 
ing body opposite two stations on the shore, especially when it is not 
convenient to stretch a string across. An arch of a bridge is a good 
station for an experiment of this kind, because it affords a very regular 
section and two fixed points of observation ; and in some instances 
the sea practice of heaving the log may be advantageous. Where a 
timepiece is not at hand, the observer may easily construct a quarter- 
second or other pendulum, by means of the rules and table relating 
to pendulums in the Dynamics. 

2. M. Pitot invented a stream measurer of a simple construc- 
tion, by means of which the velocity of any part of a stream may 
readily be found. This instrument is composed of two long tabes of"" 
glass, open at both ends; one of these tubes is cylindrical throughout; z 
the other has one of its extremities bent into nearly a right angle, ^ 
and gradually enlarges like a funnel, or the mouth of a trumpet:^ 
these tubes are both fixed in grooTes in a triangular prism of wood,^ 
so that their lower extremities are both on the same level, standings 
thus one by the side of the other, and tolerably well preserred from^ 
accidents. The frame in which these tubes stand is graduated, closed 
by the side of them, into dirisions of inches and lines. 

To use this instrument, plunge it perpendicularly into the watery 
in such manner that the opening of the funnel at the bottom of ontfS 
of the tubes shall be completely opposed to the direction of the cur— - 
rent, and the water pass freely through the funnel up into the tube-i^ 
Then observe to what height the water rises in each tube, and note^ 



CHAP, v.] 8TRBAM MBA8UREB8. 275 

the difference of the sides, for this difference will be the height due 
to the velocity of the stream. It is manifest, that the water in the 
cylindrical tuhe will be raised to the same height as the surface of 
the stream by the hydrostatic pressure, while the water entering 
from the current by the funnel into the other tube will be compelled 
to rise aboTe that surface by a space at which it will be sustained by 
the impulse of the moring fluid: that is, the momentum of the stream 
iriU be in equilibrio with the column of water sustained in one tube 
fhore the surface of that in the other. In estimating the velocity by 
means of this instrument, we must have recourse to theory, as cor- 
rected by experiments. Thus, if A, the height of the column sus- 
tained by the stream, or the difference of heights in the two tubes, be 
in feet, we shall have e = 6'5 v^ A, nearly, the velocity, per second, 
of the stream; if A be in inches, then o = 22*47 %/ ^i nearly: or 
&rther experiments made with the same instrument may a little 
modify these coefficients. 

It win be easy to put the funnel into the most rapid part of the 
stream, if it be moved about to different places until the difference 
of altitade in the two tubes becomes the greatest. In some cases it 
will happen, that the immersion of the instrument will produce a 
little eddy in the water, and thus disturb the accuracy of the observ- 
«tioii : but keeping the instrument immersed only a few seconds 
will correct this. The wind would also affect the accuracy of the 
experiments ; it is, therefore, advisable to make them where there is 
little or no wind. By means of this instrument a great number 
of curious and useful observations may easily be made: the velo- 
city of wator at various depths in a canal or river may be found 
with tolerable accuracy, and a mean of the whole drawn, or they 
maj be applied to the correcting of the theory of waters running 
down gentle slopes. The observations may likewise be applied to 
nmeertBon whether the augmentations of the velocities are in propor- 
tion to the increase of water passing along the same canal, or what 
other relations subsist between them, &c. 

Where great accuracy is not required, the tube with the funnel 
%t hottom will alone be sufficient, as the surface of the water will be 
Indicated with tolerable precision bv that part of the prismatic 
fimme for the tube which has been moistened by the immersion : and 
the wdocUks may be marked against those altitudes in the tube which 
indicate them. 

d. Another good and simple method of measuring the velocity of 
water in a canal, river, &c., is that described by the Abbe Mann^ in 
Itis treaitise on rivers; it is this: — Take a cylindrical piece of dry light 
^wood, and of a length something less than the depth of the water in 
the river ; about one end of it let there be suspended as many small 
weights as may keep the cylinder in a vertical or upright position, 
with ita head^just above water. To the centre of this end fix a 
•mall straight rod, precisely in the direction of the cylinder's axis, in 
order that, when the instrument is suspended in the water, the devia* 

T 9 



276 STRSAM MKA8UREBS. [l 

tions of the rod from a perpendicularity to the mvrfmce of 
indicate which end of the cylinder goes foremost, by wbieh 
discovered the different velocities of the water at different 
for when the rod inclines forward, according to the direction 
current, it is a proof that the surface of the water has the 
velocity ; but when it reclines backward, it shows that the 
current is at the bottom ; and when it remains perpendiculai 
sign that the velocities at the top and bottom are equal. 

This instrument, being placed in the current of a river o 
receives all the percussions of the water throughout the whol 
and will have an equal velocity with that of the whole carre 
the surface to the bottom at the place where it is pat : 
by that means may be found, both with exactness and e 
mean velocity of that part of the river for any determinate 
and time. 

But, to obtain the mean velocity of the whole section of tl 
the instrument must be put successively both in the mid 
towards the sides, because the velocities at those places are of 
different from each other. Having by these means found the 
velocities, from the spaces run over in certain times, the aritl 
Tuean proportional of all these trials, which is found by divi< 
common sum of them all by the number of the trials, will 
mean velocity of the river or canal. And if this medium vel 
multiplied by the area of the transverse section of the waten 
place, the product will be the quantity running through that ; 
a second of time. 

The cylinder may be easily guided into that part which ^ 
to measure, by means of two threads or small cords, wh 
persons, one on each side of the canal or river, must hold and 
taking care at the same time neither to retard nor accelei 
motion of the instrument. 

4. Let A A' B B' be a hollow cylinder, open at both ends, 
it be capable of being fixed by the side of a platform or of a 
that its lower extremity BB^ may be placed at 
any proposed depth below H R, the upper surface 
of the stream. Let P P^ be pulleys, fixed at 
opposite sides of the top and bottom of the tube. 
To O, a globe of specific gravity nearly the same 
as that of water, let a cord OP' PS be attached, 
passing freely over the pulleys P', P, and having 
sufficient length towards S to allow of its running 
off to any convenient distance. Then, the bottom 
of the tube being immersed to any proposed 
depth, let the globe G be exposed to the free operation 
stream; and as it is carried along with it, it will in 1, 2, i 
seconds, or any other interval of time, drew off from a fizei 
as S, a portion of cord; from which and the time elapsed, the 
at the assigned depth will become known. 




] STBBAM MBA8URBB8. 277 



mvented, in 1720, an instrument called the Marine 
'y for the doable purpose of measuring a ship's way, and 
ing the velocities of streams. It is described in the Phil. 
oL dd ; and in the succeeding volume a curious example of 
given in ^ tables showing the strength and ffradual increase 
aw of the tides of flood and ebb in the nver Thames, as 
in Lambeth Reach." They are too extensive to be inserted 
are very interesting, and may be seen in the Philosophical 
ms AMdptedy vol. vii. p. 133. 



378 EQUILIBRIUM OF ILASTIC FLUIDS. [PABT II. 



CHAP. VI. 

PNEUMATICS. 

Sect. I. Weight and Equilibrium of Air and EUutic Fluids. 

1. The fundamental propositions that belong to hydrostatics are 
common to the compressible and incompressible fluids, and need not, 
therefore, be repeated here. 

2. Atmospheric air is the best known of the elastic fluids, and has ^ 
been defined to be an elastic fluid, having weight, and resisting com*^ ^ 
pression with forces that are directly as its density, of inversely as ^ 
the spaces within which the same quantity of it is contained. 

The correctness of this definition is confirmed by experiment. 

3. The weight of air is known from the Torricellian experiment, ^^^ 
or that of the barometer. The air presses on the orifice of the ^^. 
inverted tube with a force just equal to the weight of the column of ~% ^ 
mercury sustained in it. 

The weight of a cubic foot of air, with the barometer at 30 inches, ^ -r^ 
and the temperature ^5^ of Fahr., is 1*2 ounces avoirdupois, or-^^3 
about v^rd part of the weight of a cubic foot of water under the — ^ i< 
same circumstances. 

A bottle, weighed when filled with air, is found heavier than after^r ^i 
the air is extracted. The mean pressure of the atmosphere at^ -^t 
London is 14*18 lbs. on every square inch of the earth's surface^p^^i 
which is equivalent to the pressure of a column of quicksilver 28*89^^^ ^ 
inches in height; or a column of water nearly 33 feet in height— =^^* 
Hence the total pressure on the convex surface of the earth amount^^ -^^ 
to 10,686,000,000 hundreds of millions of pounds. 

The elastic force of the air is proved, by simply inverting a vesseT ^^*] 
full of air in water : the resistance it offers to farther immersion, an^^ -•^ 
the height to which the water ascends within it, in proportioira^^ 
as it is farther immersed, are proofs of the elasticity of the aixi^'C'^ 
contained in it*. 



* It is in virtue of this property, and ought to be known as ottensivdy i 
possible, that a man*s hat will serve in most cases as a temnormry life-preMrva 
to persons in hazard of drowning, by attending to the following direcdoos :- 
When a person finds himself in, or about to be in, the water, let him lay hok^ ^^ 
of his hat between his hands, laying the crown dose under his chin, and th^^ -^ 
mouth under the water. By this means, the quantity of air conuined in th^' ^ 
cavity of the hat will be sufficient to keep the head above water for icvtii^^ ' 
hoars, or until assistance can be rendered. 



i 



CHAP. VI.] WBIOHT AND EQUILIBRIUM OP ELASTIC FLUIDS. 279 

4. When air is confined in a bent tube, and loaded with diflferent 
weights of mercury, the spaces it is compressed into are found to be 
inversely as those weights. But those weights are the measures of 
the elasticity; therefore the elasticities are inversely as the spaces 
which the air occupies. 

The densities are also inversely as those spaces; therefore the 
elasticity of air is directly as its density. This law was first proved 
by Manotte's experiments. ' 

In all this, the temperature is supposed to remain unchanged.-^ 
These properties seem to be common to all elastic fluids. 

Air resists compression equally in all directions. No limit can be 
assigned to the space which a given quantity of air would occupy if 
all compression were removed. 

5. In ascending from the surface of the earth, the density of the 
air necessarily diminishes : for each stratum of air is compressed only 
by the weight of those above it ; the upper strata are therefore less 
compressed, and of course less dense than those below them. 

6. Supposing the same temperature to be difl^used through the 
atmosphere, if the heights from the surface be taken increasing in 
arithmetical progression, the densities of the strata of air will decrease 
in geometrical progression. Also, since the densities are as the com- 
pressing forces, that is, as the columns of mercury in the barometer, 
the heishts from the surface being taken in arithmetical progression, 
the columns of mercury in the barometer at those heignts will 
decrease in geometrical progression. 

7. Logarithms have, relatively to the numbers which they re- 
present, the same property, therefore if b be the column of mer- 
cury in the barometer at the surface, and at any height h above 
the surface, taking m for a constant coefficient, to be determined 
by experiment, 

A = in(log6 — logg), or A = mlog- (I.) 

where m may be determined by finding trigonoroetrically the value 
of A in any case, where b and $ have been already ascertained. 

8. If ^ be the height of the mercury in the barometer at the 
lowest station, fi at the highest, t and f the temperatures of the air at 
those stations, f the fixed temperature at which no correction is 
required for the temperature of the air; and if g and / be the 
temperatures of the quicksilver in the two barometers, and n the ex- 
pansion of a column of quicksilver, of which the length is 1, for each 
degree of heat; k being the perpendicular height (in fathoms) of the 
one station above the other, 

il = 10000(1 + -00244/^'-^ -A log —^ 77^] (II.) 

« being nearly = j^. 



280 ATMOSPHBRIO ALTlTUDli. [PABT 11. 

If the oentigrade thermometer it used, hecaase the hegtHning of 
the scale agrees with the temperature f^ so that /*« 0» the formula 
hecomes more simple ; and if the expansion for air and nercurj he 
hoth adapted to the degrees of thitf scale, 

*= ,0000{. + 0044l(l±i:) log __J__^} (HI.) 

9< The temperature of the air diminishes on ascending into the 
Ittmosphere, hoth on account of the greater distance from the earth, 
the principal source of its heat, and the greater power of aheorbing 
heat that air acquires by being less compressed. 

10. Professor Leslie, in the notes on his EkmenU qf Geometry^ 
p. 495 (edit. 2nd), has given a formula for determining the tempera- 
ture of any stratum of air when the height of the mercury in the 
barometer is given. Tlie column of mercury at the lower of two 
stations being &, and at the upper /?, the diminution of heat, in 

degrees of the centigrade thermometer, is f - — T f 2^* Which 

seems to agree well with observation. 

1 1 . If the atmosphere were reduced to a body of the same density 
which it has at the surface of the earth, and of the same temperature, 
the height to which this homogeneous atmosphere would extend is, 

in fathoms, equal to 4343 ^1 + '00441 ——V or, taking the 

expansion according to Laplace, = 4343 ( 1 + ttt^ ) • • • C^^*) 

Hence if h be the height of the mercury in the barometer, reduced 

to the temperature /, the specific gravity of mercury is to that of airi 

as ft to 4343 ( I + Tjwjj: J ^ or the specific gravity of air 

h 



72 X 4343 X (\ ^ ~\ ' 
\ ^ 1000/ 

The divisor 72 is introduced in consequence of b being expressed 
in inches.— (jP/tfj^atV^ Outiines.) 

12. The^ Telocity with which air rushes into a vacuum is equal 
to that which a heavy body would acquire in Mling from a height 
equal to that of a homogeneous atmosphere equivalent in wemit 
to that of the air at the time. Thus if H be the height of Uie 
homogeneous atmosphere, deduced from the formula (IV.) abote, 
expressed in feet, and V the velocity of the air in feet per second^ 
Wd have from the formula, page 215, 

V = S- <V.) 



CHAP. VI.] ATM08PHBBIC ALTITUDES. 281 

From wbieh we find the value of V for the mean temperature and 
preasare to be 1860 feet per second. 

13. The velocity with which sound traverses the atmosphere de- 
pends not only upon the direction and force of the wind, but also on 
the temperature and density of the air at the time, and the quantity 
of moisture which it contuns. In the following formula given by 
Mr. Galbraitb, of Edinburgh, the whole of these circumstances are 
taken into account Let v be the velocity of sound in feet per 
second, t the temperature of the air, 8 the elastic force of vapour for 
the due point, the barometric pressure, x the latitude of the place 
of observation, v the velocity of the wind, and ^ the angle which the 
direetion of the wind makes with that of the sound ; then 

»a= (102-4225 + 01103 0(l + ^i qZ. 2e ) C^^'^'''^® 
— 0-01378. cos 2 a) H- w. cos ^ (VI.) 



8bct. II. Madtine$ for BaUing Water by the Pressure of the 
Atmoephere. 

1. The term Pump is generally applied to a machine for raising 
water by means of the air's pressure. 

2. The common suction-pump consists of two hollow cylinders, 
which have the same axis, and are joined in A C (fig. 216). The lower 
is partly immersed, perpendicularly, in a spring or reservoir, and is 
called the suction-^ube; the upper the body of the pump. At AC is a 
fixed sucker containing a valve which opens upwards, and is less than 
34 feet from the surface of the water. In the body of the pump is 
a piston D made air-tight, movable by a rod and handle, and con- 
taining a valve opening upwards. And a spout G is placed at a 
greater or less distance, as convenience may require, above the greatest 
elevation of D. 

The action of this pump is as follows. Suppose the movable pis- 
ton D at its lowest depression, the cylinders free from water, and the 
air in its natural state. On raising this piston, the pressure of the 
air above it keeping its valve closed, the air in the lower cylinder A B 
forces open the valve at A C, and occupies a larger space, viz., be- 
tween B, the surface of the water, and D ; its elastic force, therefore, 
being diminished, and no longer able to sustain the pressure of the 
external air, this latter forces up a portion of the water into the cylin- 
der AB to restore the equilibrium. This continues till the piston has 
reached its greatest elevation, when the valve at A C closes. In its 
subeeqaent descent, the air below D becoming condensed, keeps the 
valve at AC closed, and escapes by forcing open that at D till the 
piston has reached its greatest depression. In the following turns a 
similar effect is produced, till at length the water rising in the cylin- 
der forces open the valve at A C, and enters the body of the pump ; 



282 PUMPS. [part II. 

when, by the descent of D, the valve in AC is kept closed, and the 
water rises through that in D, which on re-ascending carries it for- 
ward, and throws it out at the spout O. 

3. Cor, 1. The greatest height to which the water can be raised in 
the common pump by a single sucker is when the column is in eqoi- 
Hbrio with the weight of the atmosphere, that is, between 32 and 36 
feet. 

4. Cor. 2. The quantity of water discharged in a given time is de- 
termined by considering that at each stroke of the piston a quantity 
is discharged equal to a cylinder whose base is a section of the pump, 
and altitude the play of the piston. 

5. To determine the force necessary to overcome the resistance 
experienced by the piston in ascending. Let h = the height H F (fig. 
217) of the surface of the water in the body of the pump above EF, 
the level of the reservoir; and a^ = the area of the section MN. 
Let h I ^ the height of the column of water equivalent to the pres- 
sure of the atmosphere ; and suppose the piston in ascending to arrive 
at any position mn which corresponds to the height IF. It is evi- 
dent that the piston is acted upon downwards by the pressure of the 
atmosphere ^ d^ h.^ and by the pressure of the column B m = er 
X H I ; therefore the whole tendency of the piston to descend = a* 
(A + HI). 

But the piston is acted upon upwards by the pressure of the air on 
the external surface £ F of the reservoir = c^ hy\ part of which is 
destroyed by the weight of the column of water having for its base 
mn, and height FI; 

.*. the whole action upwards = a^ x (^j — FI); 

whence F = a* . (A, -f HI) = 0^.(^1— FI) 
= a«. FH == d'h, 

that is, the piston throughout its ascent is opposed by a force equal to 
the teeight of a column of water having the same hose as the piston^ 
and an altitude equal to that of the surface of the water in the bodjf of 
the pump above that in the reservoir. In order, therefore, to produce 
the upward motion of the piston, a force must be employed equal to 
that determined above, together with the weight of the piston and 
rod, and the resistance which the piston may experience in conse- 
quence of the friction against the inner surface of the tube ♦. 

* Suppose the body of the pump to be 6 inches in diameter, and the greatest 
height to which the water is raised to be 30 feet ; suppose, also, the weight of 
the niston and its rod to be 10 lbs., and the friction one-fifUi of the whole 
weight. Then, -^ of the square of the diameter gives the ale gallons in 
a yard in length of the cylinder, and an ale gallon weighs 10| Iba. Therefore 
(6< X 10) + % (6* X 10) »= 360 -h 7-4 " 367*4 lbs., weight of the opposing 
column of water. And 367 4 + 10 -f i (377*4) = 452*9 lbs., whole opposing 
pressure. 

If the piston rod be moved by a lever whose arms are as 10 to 1, this pres- 
snre will be balanced by a force of 46*29 lbs., and overcome by any greater 
force. 



CHAP. VI.] SUCTION PUMP. 383 

When the piston begins to descend, it will descend by its own 
weight ; the onhr resistance it meets with being friction, and a slight 
impact against the water. 

6. Cor. 1. If the water has not reached the piston, let its level 
be in e ir. The under surface of the piston will be pressed by the 
mtemal rarefied air. But this air, together with the oolumn of 
water, £ o, is in equilibrio with the pressure of the atmosphere a' A, ; 
and . -• its pressure =s a' . (Ai — E 9). And the pressure downwards 

= «•*.; 



Hence the force requisite to keep the piston in equilibrio increases 
as the water rises, and becomes constant and = a'A as soon as the 
water reaches the constant level B H. 

7- Cor, 2. If the weight of the piston be taken into the account, 
let this weight be equal to that of a column of water whose base is 
m II and height/), ss a*/? ; 

.-. F = (i'. (Er+/?). 

8. To determine the height to which the virater will rise after one 
stroke of the piston ; the fixed valve being placed at the junction of 
the suction-tube and body of the pump : supposing that afler eyery 
elevation of the piston there is an equilibrium between the pressure 
of the atmosphere on the surface of the water in the reservoir, and 
the elastic force of the rarefied air between the piston and surface of 
the column of water in the tube, together with the weight of that 
column. 

Let ah {j^g, 218) be the surface of the water in the suction-tube, 
after the first stroke of the piston : if the piston were for an instant 
stationary at D, the pressure of the atmosphere would balance E 6, 
and the elastic force of the air in N a. 

Let A E, the height of the suction-tube, = a, 
D R, the play of the piston, == &, 

h s: the height of a column of water equivalent to the pressure of 
the atmosphere, 

y ^ the height of a column equivalent to the pressure of the air 
in No, 

m =E Eo, 

^ =r 3*14159 

and B and r = the radii of the body and the suction tube. 

Then « + jr = A, 

AF vHa hf^a 

I hr'* a 
whence A = ^0 + ^- , . , -r- ; 

.-. AR«6 + hf*a — hr*a = R**« + r*a« — r*a?» -f- ht^a. 



-*y^* ^ reduced vj^^^li^^rf^^ 
^.«. these arc ^^ ^^^^^^^^Tii^^^^. 



^-^ — - — TnX. 



«A BO on. ,^ " 1 ^ »8ceot8' ?fc+« —*•--" ' 






rise aftet^'/tVve «'*-*'" 



CHAF. VI.] SUCTION PUMP. 285 

Uon due to each particular stroke, the differences of those elevations, 
and the successive differences in the elastic force of the remaining air, 
may be known. 

10. If the weight of the valve c be not considered, it is evident 
that after a certain number of strokes a vacuum will be produced in 
the suction-tube, provided it be equal to or not greater than the 
height due to the pressure of the atmosphere, that is, if a be not 
greater than h. 

For, in this case, x^ =. x^.,, 

•nd.-.«^,= J{jt,— y^T* — 4Am*-4a?. . . (A -f o — «.-i)}, (IX.) 

whence «»., = ky the greatest height of the column of water in the 
tube. If, therefore, the length of the suction-tube do not exceed the 
height due to the pressure of the atmosphere, the water will continue 
to ascend in it after every stroke of the piston, till at length it will 
into the body of the pump. 

But if the altitude of A F be greater than k, the water will con- 
tinue to ascend without ever reaching its maximum height. For, in 
this case, an actual vacuum cannot be produced ; and as ^^ -f y* = ^'* 
and y, can never become = 0; .*. ^« can never = ^*. But, tlie 
successive values of ^r continually decreasing, the corresponding values 
of as will continually increase. 

11. If the weight of the valve c be taken into the account, a 
column of water must be added equal to the additional pressure to be 
overcome. Let / = the height of this column, then 

a + y^l^h; 
and .'. a + y ^ h ^ I = hy 

If therefore this value of A| be substituted for A, the preceding 
equations are applicable. 

12. In the preceding cases, the movable piston has been supposed 
to descend to A C. If it does not, it may happen that the water may 
not reach AC, though AC be less than' 34 feet from the surface of 
the water in the reservoir. 

After the first elevation of the movable piston to its greatest alti- 
tude, c being closed, the elastic force of the air between DN and AC 
is (A — «), and its magnitude x h R*. If, in descending, the piston 
describes a space ^ less than ^, so as to stop at a distance b ^ b* 
from AC, this magnitude becomes {b -- b") . v B,^ ; .*. the elastic force 

is (A — «) . Y 7T. Now in order that the pressure upwards may 

open the valve, this must exceed the elastic force of the atmosphere ; 

* Hence it appears that it is not ttrieUy true, that wat«r will ascend In the 
•action-tube to a height equal that of a column equivalent to the pressure of 
the atmoephere. This is a limit to which it approximates, but does not reach 
in a finite time. 



286 SUCTION PUMP. [part II, 





SUCTION 


PUHP. 




.-.(*-«), 


h 


or 


(A - *) . i 


>h.{h-b'); 




.:hx<Lhh', 


X V 



If .% T-be less than -, the valve DN will Dot open; there 
h 

will therefore be the same quantity of air between A C and the 
sucker: which, when the piston has reached its highest elevation, 
will have the same elastic force as that between A C and a' h' ; and 
therefore c, being equally pressed on both sides, will remain un- 
moved, and the water will not ascend. 

13. If the fixed valve be placed at the surface of the water; to 
determine the ascent of the water in the suction-tube. 

Let £ a, £ a^ be the successive heights to which the water rises ; 
then, after the first ascent of the piston, 

mo -f a — « 



whence a? =: J . {/? — -Jp^ — ^hmh} 



and y = i . {2k — p -f >//?* — 4tkmh}^ 

which equations are the same as were determined for the first ascent 
of the piston (§ 8). Therefore, in the same manner as before, 



we shall have «, = J . {/? — s/p^ — 4>hmh — 4Aa?,.,}, 

14. If the water be supposed to stop after (n -f IX ascents of the 
piston, then «, = a?»_, ; 

and .*. d?,_, = i . {/> — "^p^ — ^hmh — 4Aa?^,}, 



whence d?,_i = J . {a -f mh ± >/(a + mhY — ihmb} ...(X.) 

Hence, therefore, there are two altitudes at which the water may 
stop in its ascent, if (a + mbY is equal to or greater than 4 Am 5. In 
the former case the two values of x^^^ are equal, that is, there will 
be only one altitude = ^ . (a + m 6), at which the water will stop. 
In the latter case there are two which may be ascertained. 

If 4 Am* be greater than (a + nthf^ the water will not atop. 

Ex.l. If A = 32 feet, a = 20, 6 = 4, and m = 1, or the 
suction-tube and body of the pump be of the same diameter. 



«— I = I . {20 + 4 ± V(24)« - 4 . 1 . 32 . 4} = i . {24 ± V64} 

= le or 8. 



CHAP. VI.] VORCINO PUMP. 287 

Ex. 2. If A = 32 feet, a = 25, ft = 2, and wi = 4, 



x^. = 1 . {25 H- 8 ± >/(33)' — 4 . 32 . 4 . 2} = J . {33 ± 
>/65} = 41-8062 or 241938. 

15. If m = 1, or the tubes have the same diameter, 

«-. = i . {« + ft ± >/(« + hf - 4A6}, 
which is imaginary, if (a + 5)^ is less than 4 A ft, or ft greater than 
(o + fty 
4A ' 

In order, therefore, that this pnmp may produce its effect, the play 
of the piston muit be greater than the sqtiare of its greatest akitttde 
above Vie surface of the water in the reservoir divided by four times 
the height due to the pressure of the atmosphere, 

16. The lifing-pump consists of a hollow cylinder, the body of 
which is immersed in the reservoir. It is furnished with a movable 
piston, which, entering below, lifts the water up, and is movable by 
means of a frame which is made to ascend and descend by a handle. 
The piston is furnished with a valve opening upwards. A little 
below the surface of the water is a fixed sucker with a valve opening 
upwards. This is an inconvenient construction, upon the peculiari- 
ties of which we need not dwell. 

17- The forein^'pump consists of a suction-tube AEFC (fig. 219) 
partly immersed m the reservoir, of the body of the pump ABOC, 
and of the ascending tube HK. The body is furnished with a 
movable solid sucker or plunger, D, made air-tight. And at AC 
and U are fixed suckers with valves opening upwardd. 

18. To explain the action of this pnmp. Suppose the plunger D 
at its greatest depression ; the valves closed, and the air in its natural 
state. Upon the ascent of D, the air in A CD occupying a greater 
space, its elasticity will be diminished, and consequently the greater 
elasticity of the air in A F will open the valve at A C, whilst that at 
H is kept closed by the elasticity of the external air ; water there- 
fore will rise In the suction -tube. On the descent of D from its 
greatest elevation, the elasticity of the air in the body of the pump 
will keep the valve A C closed, and open that at H, whence air will 
escape. By subsequent ascents of the piston, the air will be ex- 
pelled, and water rise into the body. The descending piston will 
then press the water through the valve at H, which will close, and 
prevent its return into the bodv of the pump ; D therefore ascending 
again, the sp^ce left void will be filled by water pressing through the 
valve AC; and this upon the next ascent of D is forced into the 
ascending tube ; and thus, by the ascents and descents of D, water 
may be raised to the required height. 

19. In this pump D must not ascend higher than about 32 feet 
from the surface of the water in the reservoir. 

20. To determme the force necessary to overcome the resistance 
experienced by the piston : — 



288 FIRE INOINB. [past II. 

Let h = the height of a column of water equivalent to the presiare 
of the atmosphere, and E B the height to which the water is forced. 
Let M N he any position of the piston D whose area = A, and the 
weight of the piston and its appendages = P. Let X = the force 
necessary to push the piston upwards during the suction, friction not 
heing considered, and V s= that employed to force it down. 
When the piston ascends, and H is closed 
X = P + AA — A.(A — ME) 
= P -f A . ME. 

Let the sucker he in the same position in its descent, and therefore 
A C closed, and H open, 

Y = AA -fA.MB — (AA -f P) 

= A . M B — P. 

Hence X + Y = A.EB; or the whole force exerted, in the case 
of equilihrium, is equal to the weight of a column of water whose 
base is equal to that of the piston, and altitude the distance between 
the surface of the water and the point to which it is to be raised. 

21. In this pump the effort is divided into two parts, one opposed 
to the suction, and the other to the forcing ; whereby an advantage 
is gained over the other pumps where the whole force is exerted at 
once whilst the water is raised. 

22. In order to have the force applied uniform, lot X = Y ; 

.-. P-|-A.ME==A.MB — P; 

.-. P = JA . (MB — ME). 

The piston therefore must play in such a manner that M B may be - 
greater than M E. 

23. In the common forcing pump, the stream b intermitting ; for ' 
there is no force impelling it during the return of the sucker. 

One mode of remedying this, is by making an interruption in the 
ascending tube, which is surrounded by an air vessel T (fig. 220); 
in which, when the water has risen above Z, the air above it is com- 
pcssed, and by its elasticity forces the water up through Z; the 
orifice of which is narrower than that of the tube, and therefore the 
quantity of water introduced during the descent of the piston will 
supply its discharge for the whole time of the stroke, producing a 
continued stream. 

24. The fire-engine consists of a large receiver A B C D, called the 
air-vessel, into which water is driven by two forcing-pumps EF, 
OH (fig. 221), (whose pistons are Q an^ R), communicating with 
its lower extremities at I and K, through two valves opening 
inii-ards. From the receiver proceeds a tube M L through which 
the water is thrown, and directed to any point by means of a pipe 
movable about the extremity L. The pumps are worked by a 
lever, so that whilst one piston descends the other ascendi. The 
pumps communicate with a reservoir of water at N. 



CHAP. VI.] QUICKSILVER PUMP. 289 

25. To explain the action of this engine. 

The tube N being immersed iu the reservoir, and the piston R 
drawn up, the pump OH becomes filled; and the descent of the 
piston R will, as in the forcing-pump (§ 1 8), keep the valve H close, 
and cause the water to pass into the air-vessel by the valve I, whilst, 
by the weight of the water in the air-vessel, the valve K will be 
kept abut. In the same manner, when R ascends, Q descending 
will force the water through K into the air-vessel. By this means 
the air above the surface of the water becoming greatly compresse<1 
will, by its elasticity, force the water to ascend through M L, and to 
issue with a great velocity from the pipe*. 

26, When the air-vessel is half full of water, the air being then 
compressed into half its natural space will have an elastic force 
equal to twice the pressure of the atmosphere : therefore, when the 
stop-oock is turned, the air within pressing on the subjacent water 
with twice the force of the external air, will cause the water to spout 
from the engine to the height of (2 — 1)83, or 33 feet; except so 
far as it is diminished by friction. 

Or, genemlly, if denote the fractional height of the water in 

II 

the air-barrel, then - will denote the height of the space occupied by 

the compressed air, n times the pressure of the atmosphere its elastic 
force, and (» — 1 ) 33, the height in feet to which the water may be 
projected. 

Thus, if I of the air-barrel be the height of the water, the elastic 
force of the air will be four times the pressure of the atmosphere, 
and (4 — 1) 33 =s 99 feet, the height to which the water may then 
be thrown by the engine. 

27- The modifications in the constructions of pumps with a view 
to their practical applications are very numerous. Those who wish 
to acquaint themselves with some of the most useful, may consult 
the 2nd Tolume of my Treatise on MechanicSy and Nos. 13, 41, 69, 
and 93, of the Meehaniee Magazine, 

Id addition to these, there may now be presented a short account of 
uauieksiiver-vump^ which has been recently invented by Mr. Thomas 
Clark of Edmburgh, and which works almost without friction. It 
has great power in dra\nng and forcing water to any height, and is 
extremely simple in its construction. In fig. 222, a a is the main 
pipe inserted into the well b ; a valve is situated at c, and another at 
dj both opening upwards ; a piece of iron tube is then bent into a 
drcular form, as at/^ again turned off at g in an angular direction, so 
as to pass through a stuffing box at A, and from thence bent out- 
wards as at t, connecting itself with the ring. A quantity of quick- 

* Tbe preceding part of this section Is taken from Bland*s Hydrottatics ; a 
nxj dqgant and valuable work, which I beg most cordially to recommend 
to those who wish to obtain a oomprehenaive knowledge of the Uieory of this 
-■ ■ — at of adeaee. 



290 SPIRAL PUMP. [part II. 

silver is then put into tlie ring filling it from q to q\ and the ring 
being made to vibrate upon its axis hy a vacuum is soon effected 
in the main pipe by the recession of the mercury from p to q^ 
thereby causing the water to rise and fill the vacuum : upon the 
motion being reversed, the quicksilver slides back to ^, forces up the 
water and expels it at the spout e. 

*' Mr. Clark calculates that a pump of this description with a ring 
twelve feet in diameter, Hdll raise water the same height as the 
common lifting pump, and force it one hundred and fifty feet higher 
without any friction." {Mechanics' Register and Jamiesons Edin- 
burgh Journal,) 

28. It is usual to class with pumps, the machine known by the s^ 
name o^ Archimedes* scretCy or the teater-stiail. This consists either -im-t 
of a pipe wound spirally round a cylinder, or of one or more spiral J'.m^ 
excavations formed by means of spiral projections from an internal X^^a^ 
cylinder, covered by an external cylindrical case, so as to be wator-^- -r3cr 
tight. The cylinder which carries the spiral is placed aslant, so as io^^M ^\o 
be inclined to the horizon in an angle of from 30^ to 45°, and£» M-nd 
capable of turning upon pivots in the direction of its axis posited atirwEx at 
each extremity. The lower end of the spiral canal being immersedC» -^^ed 
in the river or reservoir from which water is to be raised, th^ MrMiie 
water descends at first in the said canal solely by its gravity; but «l.» *ut 
the cylinder being turned, by human or other energy, the water '^^ -^er 
moves on in the canal, and at length it issues at the upper extremitj^-^ ^^7 
of the tube. 

Several circumstances tend to make this instrument imperfect ana» ^rsi^ 
inefficacious in its operation. The adjustments necessary to insure ^* ^ 
a maximum of effectual work are often difficult to accomplish. 1 ^^ " 
seldom happens, therefore, that the measure of the work done ex— :^^" 
ceeds a tliird of the power employed : so that this apparatus, not— -^^'* 
withstanding its apjmrent ingenuity and simplicity, is very sj>aringljC— -^ 
introduced by our civil engineers. 

29. Spiral pump. This machine is formed by a spiral pipe or ^' 
several convolutions, arranged either in a single plane, as in fig. 223. > 
or upon a cylindrical or conical surface, and revolving round an axis ^ * 

The curved pipe is connected at its inner end, by a central water 

tight joint, to an ascending pipe, r P, while the other end, S, receivec. .^ 
during each revolution, nearly equal quantities of air and water««i^ 
This apparatus is usually called the Zurich machine, because it wa^ 
invented, about 1746, by Andrew Wirtz^ an inhabitant of Zurich*^ 

It has been employed with great success at Florence, and in Russia^ 
and the late Dr. Thomas Young states, that he employed it advantage— 
ously for raising water to a height of forty feet. The outer end of 
the pipe is furnished with a spoon, S, which contains as much water 
as will half fill one of its coils. The water enters the pipe a little before 
the spoon has reached its highest position, the other half remaining 
full of air. This air communicates the pressure of the column of 
water to the preceding portion; and in this manner the effect of 
nearly all the water in the wheel is united, and becomes capable of 



CHAP. VI.] SCHEMNITZ VESSELS. 291 

sapporting tlie column of water, or of water mixed with air, in the 
ascending pipe. The air nearest the joint is compressed into a space 
much smaller than that which it occapied at its entrance ; so that, 
when the height is considerahle, it becomes advisable to admit a 
larger portion of air than would naturally fill half the coil. This, 
while it lessens the quantity of water raised, lessens also the force 
requisite to turn the machine. The loss of power, supposing the 
machine well constructed, arises only from the friction of the water 
against the side of the pipes, and that of the wheel on its axis : and 
where a large quantity is to be raised to a moderate height, both of 
these sources of resistance may be rendered very inconsiderable. 

30. Schemnitz vessels^ or the Hungarian machine. The media- 
tion of a portion of air is employed for raising water, not only in the 
spird pump, but also in the air-vessels of Schemnitz, as shown in 
fig. 224. A column of water, descending through a pipe, C, into a 
closed reservoir, B, containing air, obliges the air to act, by means of 
a pipe, D, leading from the upper part of the air-vessel, or reservoir, 
on the water in a second reservoir. A, at any distance either above it 
or below it, and forces this water to ascend through a third pipe, E, 
to auj height less than that of the first column. The air-vessel is 
then emptied, the second reservoir filled, and the whole operation 
repeated. The air, however, must acquire a density equivalent to 
tlie requisite pressure before it can begin to act : so that, if the height 
of the columns were thirty-four feet, it must be reduced to half its 
natural space before any water could be raised, and thus half of the 
force would be lost. But where the height is small, the height lost 
in this manner is not greater than what is usually spent in overcoming 
friction, and other imperfections of the machinery employed. The 
force of the tide, or of a river rising and falling with the tide, might 
easily be applied to the purpose of raising water by a machine of this 
kind. Thus, if at low tide the vessel A were filled with air, then, at 
high tide, the water flowing down the tube £, would cause the water 
in the vessel B to ascend in the pipe C. 

31. The hydraulic ram. In this hydraulic arrangement, the 
momeiitum of a stream of water flowing through a lon^ pipe is 
employed to raise a small quantity of water to a considerable heiglit. 
The passage of the pipe being stopped by a valve which is raised by 
the stream, as soon as its motion becomes sufiiciently rapid, the 
whole column of fluid must necessarily concentrate its action almost 
instantaneously upon the valve. In these circumstances it may be 
regarded as losing the characteristic property of hydraulic pressure, 
and to act almost as though it were a single solid : so that, supposing 
the pipe to be perfectly elastic and inextensible, the impulse may 
overcome almost any pressure that may be opposed to it. If another 
valve opens into a pipe leading to an air-vessel, a certain quantity of 
the water will be forced in, so as to condense the air, more or less 
rapidly, to the degree that may be required for raising a portion of 
the water contuned in it to a given height. 

The late Mr. Whitehurst appears to have been the first who 

u 2 



292 FORCE OP THR WIND. [PART II. 

employed this method: it was afterwards improved by Mr. Boulton. 
But, like many English inyentione, it never was adequately estimated, 
until it was brought into public notice by a Frenchman. M. Mont- 
golfier, its re-inventor, gave to it the name wliich it now bears of the 
Hjfdravlic Ram, in allusion to the battering ram. 

The essential parts of this machine are represented in figure 225. 
When the water in the pipe A B (moving in the direction of the 
arrows) has acquired sufficient velocity, it raises the valve B, which 
immediately sto])s its farther passage. The momentum which the 
water has acquired then forces a portion of it through the valve, C, 
into the air-vessel, D. The condensed air in the upper part o( D 
causes the water to rise into the pipe £, as long as the effect of the <^^ ^e 
horizontal column continues. When the water becomes quiescent, ^..^.t 
the valve B will open again by its own weight, and the current along ^^ «-i| 
A B will be renewed, until it acquires force enougli to shut the said£» M^i 
valve B, open C, and repeat the operation. 

The motion in the horizontal tube arises from the acceleration oft^ <:^ o 
the velocity of a liquid mass falling down another tube, and com^ .fliiKim 
municating with this. 

In an experiment made upon an hydraulic ram, at Avilly, n eii- m^ -^jak 
Sen lis, by M. Turquet, bleacher, the expense of power was found t*,:^ U 
be to the produce, as 100 to 62. In anotlier, as 100 to 55; in tvr^-^^^wi 
others, as 1 00 to 57. So that a hydraulic ram placed in favourabi f ^Jblc 
circumstances, may be reckoned to employ usefully rather more thi mwM^Mn 
half its force. 

*i^* For more full accounts of the three last contrivanoes, tlr:^.JRlie 
reader may consuH the 2nd volume of my Mechanics, 



Sect. III. Force of the Wind. 

1 . Air, when in continuous motion in one direction, becomes ^^ss a 
very useful agent of machinery, of greater or less energy, accordir .^iog 
to the velocity with which it moves. Were it not for its vsrialMlfir Mlitj 
in direction and force, and the consequent fluctuations in its auppT ^ "^ly? 
scarcely any more appropriate fii'st mover could generally be wnijl^ ^""^ 
for. And even with all its irregularity, it is still so useful as to 
require a separate consideration. 

2. The force with which air strikes against a moving surfaoe, 
with which the wind strikes against a quiescent surface, is nearly 
the square of the velocity : or, more correctly, the exponent of i 
velocity, determined according to the rule siven in Example 
page 96, varies between 2*03 and 2*05; so Uiat, inmost practical/ 
cases, the exponent 2, or that of the square, may be employed wiC^ 
out fear of error. U $ he the angle of incidence, s the surface struct 
in square feet, and v the velocity of the wind, in feet, par teoond; 
then, if / equals the force in avoirdupois pounds, either of die two 
following approximations may be used, viz. : — 




CHAP. VI.] FORCB OP THE WIND. 



293 



/= 



440 



(XI.) 



or,/= -002288 c'^wn'jg (XIT.) 

Of these, the first is usually the easiest in operation, requiring only 
two lines of short division, viz., by 40 and by 1 1 . 

If the incidence be perpendicular, sin^/3 = 1, and these become, 

/= ^ = -002288 «»« (XIII.) 



3. The table in the margin exhibits the 
Force of the wind when blowing perpen- 
dicularly upon a sur^Mse of one foot square, 
at the several velocities stated. The velo- 
city of 80 miles per hour, is that by which 
the aeronaut Gamerin was carried in his 
tiaUoon from Ranelagh to Colchester, in 
June, 1802. It was a strong and boister- 
ous wind, but did not assume the character 
of a hurricane^ although a wind with that 
velocity is so characterized in Rouse's table. 
In Mr. Green's aerial voyage from Leeds, 
in September, 1823, he travelled 43 miles 
in 18 minutes, although his balloon rose to 
the height of more than 4000 yards. 



4. Borda found by experiment, in the year 1762, that the force 
of the wind is very nearly as the square of the velocity, but he 
iSBUDed that force to be ereater than what Rouse found (as expressed 
in die above tahle) in tne ratio of 111 to 100. Borda ascertained 
also^ as was natulfal to expect, that, upon difierent surfaces with the 
same velocity, the force increased more rapidly than the surface. 
M. Valz, applying ttie method of the minimum squai^es to Borda's 
resoltSy ascertained tiiat the whole might be represented by the 
formula 

^ = 0001289 a^ + 0000030541 ar* (XIV.) 

and nearly as correctly by 

y = 0-00108 »" (XV.) 

^ representing the surface in square inches (French), and p the 
force oorrespondibg to the velocity of 10 feet per second expressed in 
French pounds* 



Velocity of the 
Wfnd. 


Perpendi- 


cular force 


Milen 


= feet 


foot in 


fa one 


inane 


•▼OlIttupOlB 

pound*. 


hour. 

1 


second. 


1-47 


•005 


8 


2-93 


•020 


3 


4-40 


•044 


4 


5-87 


•079 


5 


7-33 


•123 


10 


1407 


•402 


15 


2200 


I 107 


20 


20*34 


1-908 


25 


30*67 


3075 


30 


4401 


4-420 


35 


51 34 


0027 


40 


58-68 


7-873 


45 


00 01 


0063 


60 


7336 


12-300 


00 


88 02 


17-716 


80 


117-30 


31-400 


100 


14670 


40-200 



294 WAT£R AS A MECHANICAL AOBNT. [PAHT ^j^ 



CHAP. VII. 

MECHANICAL AGENTS. 
Sect. I. Water as a Mechanical Anient, 

1. The impulse of a current of water, and sometimes its weight 
and impulse jointly, are applied to give motion to mills for grinding 
corn and for various other purposes. Sometimes the impulse is ap- 
plied ohiiquely to float-boards in a manner which may be compre- 
hended at once by reference to a amoke-Jack^ in which the asoendifiS 
smoke strikes the vanes obliquely, and communicates a rotatory in*" 
tion. If we imagine the wLole mechanism to be inverted, m^'^*^ 
to fall upon the vanes, rotation would evidently be produced ; m^^, 
that with greater or less energy in proportion to the qoantitj ^ 
water and the height from which it falls. i 

Water-wheels of this kind give motion to mills in Germany, a- ^^ ^ 
some other parts of the Continent of Europe. I have also w""'^; 
mills of the same construction in Balta, the northernmost Sbetla^^^ 
isle. But wherever they are to be found, they indicate a very vT^^ 
perfect acquaintance with practical mechanics; as they occasion 
considerable loss of power. 

2. Water frequently gives motion to mills, by means of what 
technically denominated an undershot wheel. This has a number ^ 
planes disposed round its circumference, nearlv in the direction m ^ 
its radii, and these float-hoards (as they are called) dipping into tl*^" 
stream, are carried round by it, as shown in fig. 226. The axle c^ 
the wheel, of course, by "the intervention of proper wheels an' ^^ 
pinions, turns the machinery intended to be moved. Where tb ^^ 
stream is large and unconfined, the pressure on each float-board is thi^^ 
corresponding to the head due to the relative velocities (or differenc^^^ 
between the velocities of stream and float-board) : this pressare v^^ 
therefore, a maximum when the wheel is at rest; but the wwrkfer-^'^ 

formed is then nothing. On the other hand, the pressure is noibin^^ 
when the velocity of the wheel equals that of the stream. Goose — "^ 
quently, there is a certain intermediate velocity, which CMises tb^^ 
work performed to be a maximum. 

The weight equal to the pressure is Q (^/ H — ^ ^)', Q beinp 
the quantity of water passing in a second, H the height due to V the 
velocity of the water, and h that due to U the velocity of the floefr- 
board. Considering this as a mass attached to the wheel, its moTing 
force is obtained by multiplying it into U : and as >/ H — ^^ A vmries 



CHAP. VII.] WATBBMJLLS. 295 

as V — U, this moviDg force varies as (V — U)^ . U which is a maxi" 
mum when U = ^l V. In this case, then, the rim of the wheel moves 
with ^ of the velocity of the stream ; and the effect which it pro- 
duces is 

Q X (fV)« X 4V = ^QV»: 

80 that the work performed hy an undershot wheel, according to the 
usual theory, equals -^-j of the moving force. 

Friction, and the resistance of fluids, modify these results; hut 
Smeaton and others have found that the maximum work is always 
obtained when U is between ^ V and ^ V. 

3. Where the floats are not totally immersed, the water is heaped 
upon them ; and in this case the pressure is that due to 2 H. 

4. When the float-boards move in a circular sweep close fitted to 
them, or, in general, when the stream cannot escape >vithout ac- 
quiring the same velocity as the wheel, the circumstances on which 
the investigation turns become analogous to what happens in the 
collision of non-elastic bodies. The stream has the velocity V before 
the stroke which is reduced to U, and the quantity of motion corre- 
sponding to the difference, or to V — U, is transferred to the wheel ; 
this turns with the velocity U; and therefore the eflfcct of the wheel 

(V— U\ VU— U* ,.,. 
I U, or — ; which is a maximum when V ^ 

2 U ; being then ^ of the moving power. 

Hence appears the utility of constraining the water to move in a 
narrow channel. 

5. The undershot wheel is used where a large quantity of water 
can be obtained with a moderate fall. But where the fall is con- 
siderable the overshot (fig. 227) is almost always employed. Its cir- 
cumference is formed into angular buckets, into which the water is 
delivered eithef at the top or within 60** of it : 52** | is the most 
advantageous distance. In that case, if r = the full radius of the 
wheel, H the whole, and h the effective height of the fall, A = r 
(1 -i- sin 37*"^) = 1-605 r, and r = '623 h. If the friction be 
aboat f of the moving power, the velocity of the circumference of 
the wheel to produce a maximum effect, will = 2*07 >/ H. Here, 
too, a fall of -^ H will give the water its due velocity of impact upon 

the wheel: and 122*176 s U^ equals the mechanical effect in 
pmrndfl, 8 being the section, in feet, of the stream that supplies the 
backets. 

Mr. Smeaton's experiments led him to conclude that overshot 
wheels do most work when their circumferences move at the rate of 

3 feet in a second, and that when they move considerably slower than 
this, they become unsteady and irregular in their motion. This de- 
termination is, however, to be understood with some latitude. He 
mentions a wheel 24 feet in diameter, that seemed to produce nearly 
ils fall effect though the circumference moved at the rnte of G feet in 
a second ; and another of the diameter of 33 feet, of which the cir^ 



296 WATEBMILLS. [PABT H. 

curafereiicc had only a velocity of 2 feet in a second, without any 
considerable loss of power. The first wheel turned round in 12*6 
seconds, the latter in 51*9 seconds. 

0. Where the fall is too small for an overshot wheel, it is most 
advisable to employ a breast^wheel (Plate VI.) which partakes of iu 

{)roperties ; its float-boards meeting at an ansle, so as to be assimi- 
ated to buckets, and the water beins: considerably confined within 
them by means of an arched channel fitting moderately close, but 
not so as to produce unnecessary friction. But when the circum- — 
stances do not admit of a breast- wheel, then recourse must be had to^:=a 
the undershot. For such a wheel it is best that the float-boards be^^ 
so placed as to be perpendicular to the surface of the water at th^^ j 
time they rise out of it ; that only one half of each should ever b»^^:3 
below the surface, and that from 3 to 5 should be immersed at onc^»«^: 
The Abbe Mann proposed that there should not bo more than six c^ 
eight float-boards on the whole circumference. 

7. Mills moved by the re-action of water are usually denominate^K^ 
Barkers Mills; sometimes, however. Parent's; at others, Segner^ ^ 
But the invention is doubtless Dr. Barkers. Their construction i 
shown in fig. 228, where C D is a vertical axis, moving on a pivot 

D, and carrying the uj)per millstone m, after passing through e 
opening in the fixed millstone C. Upon this axis is fixed a vertic^^ < 
tube T T communicating with a horizontal tube A B, at the extr-^B»t 
mities of which A, B, are two apertures in opposite directions. Wh— ^ 
water from the mill-course MN is introduced into the tube TT, H 
flows out of the apertures A, B, and by the re-action or count^^'- 
pressuro of the issuing water the arm AB, and consequently {9^e 
whole machine, is put iu motion. 

In order to understand how this motion is produced, let us svp* 
pose both the apertures shut, and the tube T T filled with water up 
to T. The apertures A and B, which ore shut up^ will be presseJ 
outwards by a force equal to the weight of a column of water whose 
height is T T, and whose area is the area of the apertures. Every 
part of the tube A B sustains a similar pressure ; but as these pres- 
sures arc balanced by equal and opposite pressures, the arm A B is at 
rest. By opcnhig the aperture at A, however, the pressure at that 
place is removed, and consequently the arm is carried round bv a 
pressure equal to that of a column TT, acting upon an area equal to 
that of the apei'ture A. The same thing happens on the arm TB; 
and these two pressures drive the arm A B round in the same direc- 
tion. This machine may evidently be applied to drive any kind of 
tnachinery, by fixing a wheel upon the vertical axis CD. 

8. Mr. Runisey, an American^ and Mr. Segner, improved this 
machine, by conveying the water from the reservoir, not by a pipe, 
in greater part of which the spindle tuklis, but by a pipe which de- 
scends from a reservoir, as F, until it reaches lower than tlie arms 
A B, and then turns up by a curvilinear neck and collar, entering 
between the arms at the lower part, as shown in fig. 229. This 
greatly diminishes the friction. 



CHAP. VII.] AIB AS A MKCHANICAL AGENT. 297 

9. Professor Playfair has correctly remarked that the raoving force 
becomes greater after the machine has hegan to move ; for the water 
in the horisoDta) arms acquires a centrifugal force, hy which its pres- 
sure against the sides is increased. When the machine works to the 
greatest advantage, the center of the perforations should move with 

the TelociQr — '\/hg^ where r is the radius of the horizontal arm, 

mearared from the axis of motion to the center of the perforation, 
and r J the radius of the perpendicular tube, g being put for the force 
of gravity, or 82 j^ feet. 

As 2 V r is the circumference described by the center of each per- 
foration, — =r is the Ume of a revolution in seconds. 

The quantity — y/hg is also the velocity of the effluent water; 

therefore, when the machine is worked to the greatest advantaee, the 
velodty with which water issues is equal to that with which it is 
carried borisontallv in an opposite direction ; so that, on coming out, 
it falls perpendicularly down. 

10. The following dimensions have been successfully adopted; 
vis. radius of the arms from the center of the pivot to the center of 
the disdiarging holes, 46 inches; inside diameter of the arms, 
3 inches ; diameter of the supplying pipe, 2 inches ; and height of 
the working head of water, 21 feet above the point of discharge. 
When the machine was not loaded, and had but one orifice open, it 
made 115 turns in a minute. This agrees to a velocity of 46 feet in 
a second for the orifice, being greater than the full velocity due to the 
head of water by between 9 and 10 feet ; the difference is due to the 
effect of the centrifugal force. 

The the<»y of this machine is yet imperfect \ but there can be no 
donbt of its utility in cases where the stream is small, with a con- 
stderable fidl. 

Mr. James Whiteland, a correspondent of the Franklin Joumaiy 
proposes to make the horizontal arms of the mill of a curved form, 
BQch that the water will run from the center to the extremity of the 
arms in a straight line when the machine is working. For the me- 
thod of constructing the curve, see Mechanics Magazine^ No. 499. 
It is very clear, however, that the additional efficiency of the 
machme will not be so great, by any means, as the inventor anti- 
cipates. 



Sbct» IL Air a$ a Mechanical Agent 

In the application of wind to mills, whatever varieties there may 
W in thrir internal structure, there are certain rules and maxims vrith 



298 AIR A8 A MECHANICAL AGENT. [PABT II. 

regard to the position, shape, and magnitude of the sails, which will 
bring them into the best state for the action of the wind, and the pro- 
duction of useful effect. These haye been considered much at laige 
by Mr. Smeaton ; for this purpose he constructed a machine, of 
which a particular description is given in the Pkilosopkieal Tranm»' 
tions^ vol. 51. By means of a determinate weight it carried round an 
axis with an horizontal arm, upon which were four small movemble sails. 
Thus the sails met with a constant and equable blast of air ; and at 
they moved round, a string with a weight affixed to it was wound 
about their axis, and thus showed what kind of size or construction 
of sails answered the purpose best. With this machine a great 
number of experiments were made : the results of which were as fol- 
lows : — 

(1.) The sails set at the angle with the axis proposed as the best 
by M. Parent and others, viz. 55% was found to be the worst pro- 
portion of any that was tried. 

(2.) When the angle of the sails with the axis was incrcaiied from 
72° to 75°, the power vths augmented in the proportion of 31 to 45; 
and this is the angle most commonly in use when the sails are 
planes. 

(3.) Were nothing more requisite than to cause the sails to acquire 
a certain degree of velocity by the wind, the position recommended 
by M. Parent would be the best. But if the sails are intended, with 
given dimensions, to produce the greatest effect possible in a given 
time, we must, if planes are made use of, confine onr angle within 
the limits of 72° and 75°. 

(4.) The variation of a degree or two, when the angle is near the 
best, is but of little consequence. 

(5.) When the wind falls upon concave sails it is an advantage to 
the power of the whole, though each part separately taken tihonld not 
be disposed of to the best advantage. 

(6.) From several experiments on a large scale, Mr. Smeaton has 
found the following angles to answer as well as any. The radius is 
supposed to be divided into six parts ; and ^th, reckoning from the 
center, is called 1, the extremity being denoted 6. 

No. Angle with th??Si2^ 

1 72° IS'' 

2 71 19 

3 72 18 middle 

4 74 16 

5 771 121 

6 83 7 extremity. 

(7.) Having thus obtained the best method o£ weat^erin^ the sails, 
t. e. the most advantageous manner in which they can be placed, our 
author's next care was to try what advantage could be deriTed from 
an increase of surface upon the same radius. The resolt obtained 
was, that a broader sail requires a larger angle ; and when the shI 



(Nil proper lur ciiiiirgtfu hmia ; ii uciiig luuiiu in prauucc umi uiis 
bould rather be too little than too much exposed to the direct 

of the wind. 

nc have imagined, that the larger the sail the greater would be 
iwer of the windmill, and have therefore proposed to fill up the 

area ; and by making each sail a sector of an ellipsis, acconling 
Parent's method, to intercept tlic whole cylinder of wind, in 

to produce the greatest effect possible. From our author's 
iments, however, it appeared, that when the surface of all the 
zeeedcd seven-eighths of the area, the effect was rather dimi- 
I than augmented. Hence he concludes, that when the wliolo 
er of wind is intercepted, it cannot then produce the greatest 
for want of proper interstices to escape. 

t is certainly desirable," says Mr. Smeaton, ^' that the sails of 
nills should be as short as possible ; but it is equally desirable, 
be quantity of cloth should be the least that may be, to avoid 
>e by sudden squalls of wind. The best structure, therefore, 
"ge mills, is that where the quantity of cloth is the greatest in a 
circle that can be: on this condition, that the effect holds out 
^portion to the quantity of cloth ; for otherwise the effect can 
Rented in a given degree by a lesser increase of cloth upon a 

radiua than would be required if the cloth was increased upon 
me radius." 

) The ratios between the velocities of windmill sails unloaded, 
rhen loaded to their maximum, turned out very different in dif- 
; experiments; but the most common proportion was as 3 to 2. 
aeral it happened that where the power was greatest, whether 

enlargement of the surface of the sails, or an increased velocity 
! wind, the second term of the ratio was diminished. 
) The ratios between the least load that would stop the sails 
le maximum with which they would turn, were confined be- 

that of 10 to 8 and 10 to 9 ; being at a medium about 10 to 

Old 10 tn 0. or nhnnt A to /i; thnnrrh nn th<» whnio if orkrMMii*o«1 



300 SUSATON's bulbs fob windmills. [pAHT 11. 

mils in ten or fifteen minutes ; and, from the length of the arms from 
tip to tip, has computed, that if an hoop of the same size was to ran 
upon plain ground with an equal velocity, it would go upwards of 
thirty miles in an hour. 

(11.) The load at the maximum is nearly, but somewhat less than, 
as the square of the velocity of the wind ; die shape and position of 
the sails being the same. 

(12.) The effects of the same sails at a maximum are nearly, but 
somewhat less than, as the cubes of the velocity of the wind* 

(13.) The load of the same sails at a maximum is nearly as the ^ 
squares, and the effect as the cubes of their number of turns in a^m 
given time. 

(14.) When sails are loaded so as to produce a maximum at m^ 
given velocity, and the velocity of the wind increases, the load con— <tf 
tinuing the same; then the increase of effect, when the increase oft-^s 
the velocity of the wind is small, will be nearly as the sqnares oft^-o 
these velocities: but when the velocity of the wind is double, th^^: 
effects will be nearly as 10 to 27|; and when the velocities com— jC3 
pared are more than double of that where the given load produces i^ 
maximum, the effects increase nearly in a simple ratio of the velocit)^-' 
of the wind. Hence our author concludes, that windmills, snch wm^ 
the different species for draining water, &c, lose much of their effecft-^ 
by acting against one invariable opposition. 

(15.) In sails of a similar figure and position, the number of tium^ ^ 
in a given time will be reciprocally as the radius or length of the^ ^m 
sail. 

(IG.) The load at a maximum that sails of a similar figure ■w i j 
position will overcome, at a given distance from the center of motioo0^ 
will be as the cube of the radius. 

(17.) The effects of sails of similar position and figure are as tb« 
square of the radius. Hence augmenting the length of the sail with- 
out augmenting the quantity of cloth, does not increase the power; 
because what is gained by length of the lever is lost by the slownen 
of the motion. Hence also, if the sails are increased in length, the 
breadth remaining the same the effect will be as the radios. 

(18.) The velocity of the extremities of the Dutch sails, as well 
us of the enlarged soils, either unloaded or even when loaded to a 
maximum, is considerably greater than that of the wind itself. This 
appears plainly from the observations of Mr. Fciguson, already re- 
lated, concerning the velocity of soils. 

■ (19.) From many observations of the comparative effects of aaih 
of various kinds, Mr. Smeaton concludes, that the enlarged sails are 
superior to those of the Dutch construction. 

(20.) He also makes several just remarks upon those windmills 
which are acted upon by the direct impulse of the wind against aaila 
fixed to a vertical shaft : his objections have, we beliere, been joa* 
tified in every instance by the inferior efficacy of these horiaontal 
mills. 

^* The disadvantage of horisontal windmillsj" he remarks, ^ does 



CHAP. VII.] coulomb's EXPBRIMBNTS. 301 

not consist in this, tliat eacb sail, when directly opposed to the wind, 
is capable of a less power than an oblique one of the same dimen- 
sions ; but that in an horizontal windmill little more than one sail 
can be acting at once : whereas in the common windmill, all the four 
ad together ; and therefore, supposing each Tane of an horizontol 
windmill to be of the same sise with that of a vertical one, it is mani- 
fest that the power of a vertical mill ^nll be foar times as great as 
that of an horizontal one, let the number of vanes be what they will. 
This disadvantage arises from the nature of the thing ; but if we 
oonsider the farther disadvantage that arises from the difficulty of 
getting the sails back again against the wind, &c., we need not won- 
der if this kind of mill is in reality found to have not above one- 
eighth or one-tenth of the power of the common sort ; as has ap- 
pMred in some attempts of this kind." 

coulomb's experiments. 

M. Conlomb, whose experiments have tended to the elucidation of 
many parts of practical mechanics, devoted some time to the subject 
of windmills. The results of his labours were published in the 
Memoirs of the Paris Academy for 1781. The mills to which he 
directed bis attention, were in the vicinity of Lille, and were, in fact, 
oil mills. From the outer extremity of one sail to the corresponding 
extremity of the opposite sail, was 70 feet, the breadth of each sau 
•J feet, of which the sail-cloth when extended occupies 5j feet, 
bemg attached on one side to a very light plank ; the line of junc- 
tion of the plank and of the sail-clod), forms, on the side struck by 
the wind, an angle sensibly concave at the commencement of the 
sail, but diminishes gradually all along so as to vanish at the remoter 
extremity. The angle with the axis, at seven feet from the shafl, is 
60*, and it increases continually so as to amount to nearly 84"^ at the 
extremity. The shaft upon which the sails turn, is inclined to the 
horizon, in different angles in different mills, from H'' to 15^. 

Coulomb infers from his experiments, 

(I.) That the raUo between the space described by the wind in a 
second, and the number of turns of a sail in a minute, is nearly con- 
stant, whatever be the velocity of the wind ; the said ratio being 
about 10 to 6, or 5 to 3. 

(2.) That with a wind whose velocity is 21^ feet per second, the 
quantity of action produced by the impulsion of the wind is equiva- 
lent to a weight of 1080 pounds avoirdupois raised 270 feet in a 
minate; the useful effect being equivalent to a weight of 1080 
pounds raised 232 feet in the same time : whence it results that the 
quantity of effect absorbed by the stroke of the stampers, the fric- 
tion, &c., is neariy a sixth part of the quantity of action. 

(3.) Suppose one of these mills to work eight hours in a day, 
Coolomb regards its daily useful effect as equivalent to that of 1 1 
horses working at a walking- wheel, in a path of the usual radius. 

(4.) It is obaemble, that in most windmills the velocity at the 



802 THE STEAM ENGINE. [pART II. 

extremity of the sails is greater than tbat of the ^iiid. In some 
coses, indeed, these velocities have been found in about the ratio of 
5 to 2. Now, it is evident that the impulsion of a fluid a^inst any 
surface whatever, can only produce pressure, or mechanical effect, 
when the velocity of the surface exposed to the impulse is less than 
that of the fluid; and that the pressure will be nothing when the 
velocity of the surface is equal to, or greater than that of, the fluid. 
Indeed, in the latter case, the pressure may operate agaimt the 
motion of the sails, and be injurious. It is desirable, therefore, in 
order to derive from a windmill all the effect of which it is sus- 
ceptible, so to adjust the number of the turns that the velocity of the 
extremity of the sails bhall be IcMy or, at most, equal to that of 
the wind. 

It would be highly expedient to make comparative experiments 
on windmills, with a view to the determination of that velocity of 
the extremity of the sails which corresponds with the maximum 
of effect. 

If v denote the velocity of the wind in feet per second, t tbe time 
of one revolution of the sails, A the angle of inclination of tbe sails 
to the axis, and D the distance from the shaft or axis of rotation 
the point which is not at all acted on by the wind, or beyond whictx 
the sail-cloth ought to be folded up ; then theoretical considerations 
supply the following theorem, viz. : — * 

D = '1092 ft? tan A. 

Ex. Suppose 9 = 30 feet per second, t =2*25 seconds, an</ 
A = 75°; then 

D = -1092 X 30 X 2-25 x 373205 = 27-509 feet 

This result agrees nearly with one of Coulomb's experiments, in 
which the velocity of the wind was 30 feet per second, the fiails 
made 17 turns in a minute, and they were obliged to fold up more 
than 6 feet from the extremity of each sail, which were 34 feet 
long, to obtain a maximum of effect. The angle A at that distance 
from the tip of the sail was 75° or 76°. 



Sect. III. Mechanical Agents depending upon Heat: — the Steam 

Engine, 

In tbe steam engine the moving power is derived from tbe yapoiir 
produced from water, by the action of heat. Vapour is formed from 
water under ordinary circumstances at its usual temperature, wbat^ 
ever that may be; but the rapidity with which it is formed, and its 
elastic force, vary with the temperature and the pressure to which 
the water is subjected. 

In order to a proper understanding of this subject, it will be 
necessary to explain the exact meaning of a few of the terms em- 



iniiy. ine uensiiy or specinc gravuy or sieam, is, uiereiore, 
Iv as its specific volume. Tlie pressure^ tensiouy or ekutic 
f the steam, is the force expressed in pounds which it exerts 
every square inch of the interior surface of any vessel con- 
it. 

en water contained in an open vessel is exposed to any source 
, its temperature becomes elevated, and a portion of the water 
erted into vapour and passes otf in that form ; this elevation 
les until the temperature (as indicated by the thermometer) 
! 212% when the vapour is formed with such rapidity as to 
e a considerable commotion in the water, which is then famili- 
id to boil. Notwithstanding, however, that fresh quantities of 
« (as before) being poured into the water, no further elevation 
emperature above 212^ can be produced so long as the vessel 
8 open. But, if we now close the vessel, so as to prevent the 
from escaping as it is formed, the sensible temperature of the 
will again begin to rise, and will continue to do so as long as 
It is applied ; it will also be found that the temperature of the 
in the upper part of the vessel will always be the same as that 
water : and further, that as its temperature increases, so will 
*ic force, or the effort which it makes to escape, also increase ; 
g found that steam contained in a closed vessel in contact 
vater and exposed to any given temperature will, under all 
stances, be of the same density and have the same pressure, 
lowledge of the nature of heat, of the precise mode in which 
"mtes in the production of steam from water, of the species of 
lation between the heat and the particles of the water, and of 
uige (if any) which the particles of the water undergo in their 
sion into steam, is so imperfect, that we are not able, by any 
JdouB or reasoning, to arrive at the law which subsists between 
isible temperature and the pressure of steam in contact with 
; as, however, the determination of the relation which they 

■\ ^€uA\ tf\f)iAti ia Aoeonfial f/k orrivini* of nrsTrt^nt rAonUo in /\tii* 



304 



THB STEAM BNOINB. 



series, are the following, which have been collected bj 
Pambour. 



Mil 

III 



Author. 



Value of p = the preuuxe 
In lbs. per square inch. 



Value of f = the 1 
bjrFahrenl 



Below 1 
atmofphere 



From 1 to 4 



From 4 to 
50 



Southern 



De Pambour 



{.SflSS.} 



(III.) 

/ 98-806 -h t y 
\ 198MS / 

(V.) 
(•S69704 + '0060091 1)* 



(11.) 
15$-72S6 (p - -OiMi 

(IV.) 
198*062 ji* - 

(VI.) 
14««1 p* - 



As these formula are of universal importance in all in 
nected with steam, whether as applied directly to the bU 
or to other purposes, such as warming buildings, &c., anc 
peculiar form having fractional or very high powers and 
comes necessary to employ logarithms in their applicatic 
express them logarithmically ; and, for the convenience o 
are not conversant with algebraical formulae, we shall 
reduce the logarithmic formulss to verbal rules, which we 
trate by a few examples. 

Expressed then, logarithmically, Southern's formulae be 

I. •.. Log(j» — -0494.8)=: 5-13 log(51-3 +0 — 11 

II. ... Log (t -f 51-3) = 2-1923601 + ^^^, ,^ 

5*lo 

De Pambonr*s — 

III. ... hogp = 6 log (98-806 + — 13-7873772; 



IV. ... LogC^ + 98-806) = 2-2978962 -f 



logp 



VI. ... Log (i + 39-644) = 2-1672906 + 



And those of Dulong and Arago — 
V. ... LogjD = 5 log (-269704 ± '0068031 0; 

logp 

Or reduced to verbal rules : — 

I. To find the pressure by Southerns formula. — Add 
temperature of the steam, multiply the logarithm of i 
5*13, and from the product subtract 11-2468073, the ret 
be the logarithm of a number, to which, if we add -049 
will represent the pressure in pounds per square inch. 



m.] TBS BTEAU EKQIVM. 305 

npie. Required the pressure of steam at the temperature of 

Log of (150 -f 51-3 = 201-3) = 23038438 
Moltipljring bj 5-13 

11-818718694 
SubtractiDg 11-246807300 



Logof 3-73174 = -57191 1394 

•73174 — -04948 = 3 78122 = the pressure required. 

fV Jind tke temperature hy SoutherriB formula,— From the 
\ in pouods subtract *04948, and divide the logarithm of the 
br bj 5-ldy to the quotient add 2*1923601, and the sum will 
ogarithm of a number, which, if we subtract 51*3 from it, 
fesent the temperature of the steam. 

1^. What is the temperature of steam whose pressure is 
per square inch ? 

Log of (10 — -04948 = 9-95052) = 9978458 
Dividing bj 5*13 

•1945118 
Adding 2-1923601 

Log of 243-70 = 2-3868719 
••. 243-70 — 51*3 = 192*4, the required temperature. 

To/wrf the pressure by De Pamhour 8 formula, — Add 98 806 
temperature, and 6 times the logarithm of their sum, minus 
3772, will be the logarithm of the required pressure. 

^pk. What is the pressure of steam at 247° ? 
Log of (247 + 98-806 = 345806) = 25388325 
Multiplying by 6 

15-2329950 
Subtracting. 13*7873772 

Log of 27901 (pressure required) = 1*4456178 

T^iind the temperature by De Pambour'i formula.— Divide 
■ithm of. the pressure by 6, and to the quotient add 
•t, the sum will be the logarithm of a number, which, if we 
•8*806 from it, will equal the temperature of the steam. 



806 THB 8TBAM SNQINB. [PIBT 

Example. Required the temperature of steam having a pi 
of 35 Ihs. 

Log of 35 = 1*5440680 
Dividing l>y... 6 

•2573447 
Adding 2-2978962 

Log of 359 121 = 2 5552409 

.-. 359-121 — 98-806 = 260215, the required temperature. 

V. To find the pressure hy Dulong and Arago*s formula.— H '^ 
dply the temperature hy -0068031, and to the product add -2697^ 
then 5 times the logarithm of their sum will he the logarithm of W 
pressure. 

Example. What is the pressure of steam having a temperature 
330°? 
Log of (0068031 X 330 + -269704 = 2-514727) = •4004909i 

Multiply by S 

Log of 100-566 (pressure required) = 2^02454g 

VI. To find the temperature by DuiUmg and Aragos formula 
Divide the logarithm of the pressure by 5, and to the quotient 6 
2-1672906, and the sum will be the logarithm of a number whicbi 
we subtract 39*644 from it, will equal the temperature. 

Example. What is the temperature of steam at the pressure 
120 lbs.? 

Log of 120 == 20791812 
Dividing by ... 5 

•4158362 
Adding 2-1672906 



Log of 382-937 = 2-5831268 

. - . 382-937 — 39*644 = 343-293 = the temperature iequire<3 

In any boiler, or other evaporating vessel, if the source of ft 
be quite uniform, so that exactly equcJ portions of heat are giveim 
to the water in equal intervals of time, it will be found that the t 
of increase in the sensible temperature of the w%^r will dimihi^ 
as the sensible temperature itself increases; that is, that the Uaof 
rature of the water will rise (for example) from 60'' to 70'' in A 



heat thus required to convert any given weight of water into 

diminishes as the temperature of the water increases, and is 
that the sum of the sensible temperature and the latent heat is 
I constant and equivalent to 1 170° of Fahrenheit's thermometer, 
this circumstance two important consequences follow, viz., that 
1 weight of steam, whatever may be its pressure and sensible 
rature, will really contain exactly the same amount of heat, 
ill therefore require precisely the same quantity of fuel for its 
•sion from water at 212° to steam at the given density, what- 
hat may be ; or in other words, that the same weight of fuel is 
ed to evaporate a given weight of water, and convert it into 

whether it be contained in an open vessel, or closed and sub- 
to any pressure, and this result has been amply confirmed by 
ment. The second consequence is, that if a quantity of steam 
larated from the water with which it was in contact, and then 
•d to expand into a larger space, or by pressure compressed into 
ler, 80 long as it does not lose any portion of its own heat, or 
e any fresh heat from surrounding bodies, its sensible tempera- 
ill always be precisely such as it would have been at the same 
re when in contact with the water from which it had been 
ced. And as in the steam engine, under ordinary circum- 
8, and where the usual precautionary means are adopted for 
iting loss of heat by the steam in its passage to the cylinder, 
mperature may without any practical error be supposed to suf- 

loss from those causes ; it results from the property which we 
'xplained above, that the temperature of the steam in the cylin- 

the engine and its pressure will always bear the same constant 
►n to each other that we have already shown they do while in con- 
ith the water in the boiler ; and, therefore, the formulee and 
already given may be applied to determine either the tempera- 
r pressure of the steam (one being known) in any part of its 
»B through the engine. 

len water contained in a closed vessel is subjected to the con- 
I action of heat in the manner we have already supposed, the 
ireture of the steam, together with that of the water, gradually 
«e8y and fresh quantities of water are converted into steam; 
rom both these causes that the elastic force or pressure of the 

increases with the elevation of its sensible temperature. It 
"esults, that as fresh quantities of water are converted into 
, while the space which contains it is but very slightly increased, 

X 2 



308 THE STEAM ENGINE. [PART II. 

the density of the steam must rapidly increase as the temperature 
rises, and its specific volume being inversely as its density, must 
become proportionally diminished. As the real quantity of water 
contained in the steam which passes through the engine, (and which 
depends upon the specific volume of the steam,) is one of the ne- 
cessary elements required in the calculation of the power of a steam 
engine, it becomes of importance to investigate the changes which 
steam undergoes in its density with any change in its temperature or 
elastic force. 

Let, therefore, V be the specific volume of steam at any tempera- 
ture t; then, since steam as well as all other aeriform fluids expand 
-^Q th part of their bulk at 32° for every degree that their sensible 
temperature is raised, putting V ^ for the specific volume of steam 
at 32°, we have 

V^ (f-32) 

480 

This formula supposes that no change has taken place in the 
pressure of the steam during its change in temperature, but as we 
have just shown that during all the changes which the steam under- 
goes in its passage through the engine, there is a constant relation 
between the sensible temperature of the steam and its pressure ; it 
therefore becomes necessary to determine in what way the specific 
volume of the steam depends upon its pressure. Now, when steam 
is separated from the water which produced it, and enclosed per se 
in any vessel, it then follows the same law relative to its density as 
any other aeriform fluid. This law has been already given at 
page 279, and is as follows, viz. : — that wliile the temperature re- 
mains the same, the elasticity of the steam, or its pressure against the 
sides of the vessel containing it, varies inversely as the space which 
it is made to occupy, or, directly as its density. 

Therefore, if P be put for the pressure of steam at 212% when its 
specific volume equals V, and p^ be put for its pressure, supposing 
its temperature unchanged but that its specific volume has become o, 
then, 

P : jt>j : : V : V 

P 

.-. t?=V — . 

Pi 
If, however, t^ represent the temperature which steam of the 
pressure p^ should have, and v^ represent its specific volume at that 
temperature, we shall have 

^ * 480 ^ ^ 480 ' 

then, substituting the value of v obtained above, and reducing in 
respect of r^, we obtain 

_ P 448jM, 
'"''' p,' 660 • 



CHAP. VII.] TUB STEAM ENOINS. 309 

Now, it is found that V, or the specific volume of Bteam at 212% is 
1700, that is, steam at 212° occupies just 1700 times the space that 
the water from which it was formed occupied, and in this state its 
pressure, or P, is equal to one atmosphere, that is, thirty inches of 
mercury, or 14*706 lbs. on the square inch. Therefore, substituting 
these values of V and P in the preceding formula, we obtain 

16969 4-37-879^, ^.,.,, 

Pi = ' (VII.) 

Pi 

By this formula it appears to he necessary that both the tempera- 
tore and pressure of the steam should be known, in order to arrive 
at its specific volume ; but we have already shown that the tempera- 
ture and pressure of the steam, both in the boiler and in its passage 
through the engine, preserve an invariable relation to each other, so 
that one can always be expressed in terms of the other; and we 
can therefore, by combining this formula with those already given, 
eliminate t^ altogether, and thereby obtain the value of e?. in terms 
of /?j only. But although the three sets of formulae whicn we have 
given, taken separately express this relation for a certain portion of 
the scale of pressures, with a sufficient degree of accuracy for all 
practical purposes, they will not admit of being correctly employed 
beyond the limits assigned to each. As, however, in the investiga- 
tion of the changes which take place in the pressure of the steam in 
the cylinder of the engine, it becomes essential to have some general 
formula which shall express with equal exactness this relation be- 
tween the specific volume and the pressure, for all temperatures and 
pressures wfiich can occur in the working of the engine, none of the 
foregoing formulae are sufficiently comprehensive to serve for this 
purpose; since, for example, the steam in an expansive engine may 
pass from a pressure of five or six atmospheres to that of one and a half 
to two atmospheres, which case would require the employment of 
both formulee (VI.) and (IV.), in order to discover the corresponding 
changes which had taken place in its specific volume. 

In order to remove this objection, Navier has proposed the follow- 
ing general formula, which expresses the specific volume of steam, 
in terms of its pressure only, and which, from the simplicity of its 
form, is peculiarly well adapted for the purposes of calculation 4^ ; it is 

v = ? (VIII.) 

in which v is the specific volume of the steam, having the pressure 
J} in lbs. per square foot; n and q are constants to be determined by 

* The following formula, derived from an expression for the relation 
between the pressure and temperature, given by the writer of the article on 
the Steam Engine, in the Encyclopaidia Metropolitana^ gives the specific volume 
of steam for all pressures above one atmosphere with singular exactness, but 
its complicated form prevents its being applied in the subsequent investigation. 

_ 12664-24 80861 6 
^ - p "^ p •"*"** * 



310 THE STEAM ENGINE. [PABT IT. 

experiment, and the values of which, according to De Pamhour, 
should he as follows, viz. : — For condensing engines, 

n = -00004227, and q = -000000258 ; 

and for non-condensing engines, 

n = -0001421, and q = -00000023. 

The reason of the distinction being made between condensing and 
non-K:ondensing engines, is that the first values of n and q are found 
more correct for low temperatures, and the second values for high 
temperatures. 

If we represent by S the space actually occupied by a given weight 
of steam at the pressure /?, and whose relative volume is o, and by S j 
the space which the same weight of steam will occupy at the pres- 
sure /7|, and whose relative volume is v^, we have the following pro- 
portion : — 

V : v^ :: S : Sj ; 

and substituting for v and v. their values in terms of p and p^y bs 
derived from equation (VIII.) we have 

1 1 



n + q p n + qp^ ' 

and reducing in respect of/?, we obtain 

^=:-'G +'.)-? "^•> 

Having thus explained the mechanical properties of steam, as far 
as is requisite for developing the theory of the steam engine, we 
shall conclude the subject by giving a table of the temperature and 
specific volume of steam for pressures varying from 5 lbs. on the 
square inch to 1 atmospheres ; the fourth column of which is cal- 
culated by FormulfiB (II.), (IV.) and (VI.), and the fifth colomn by 
Formula (VII.) 



CHAP. VII.] 



THB STEAM BNGINB. 



311 



TABLE 



OF THB PBB8SURB, TBMPBRATURB, AND SPECIFIC VOLUME OF 
FBOM 5 TO 150 POUNDS ON THB SQUARE INCH. 



STEAM, 



II 


II 

ll 


k 

h 






1 

M 

li 


II 


li 
h 


H 
=1 

III 


iilii 




T20 


5 


161-4 


4617 


7056 


49 


281-0 


664 




Mi 


6 


169-2 


3896 




7200 


50 


282'3 


563 




1008 


7 


1760 


3376 




7344 


51 


283-6 


643 




1152 


8 


1820 


2983 




7488 


52 


284-8 


684 




1296 


9 


187'5 


2673 




7632 


68 


286-0 


625 




1440 


10 


192-5 


2426 




7776 


64 


287-2 


615 




1584 


11 


1974 


2221 




7020 


65 


288-4 


607 




1728 


12 


201-3 


2050 




8064 


66 


289-6 


499 




1872 


13 


205-9 


1905 




8208 


67 


200-7 


490 




2016 


14 


2001 


1778 




8462 


68 


291-9 


483 


1 


2160 


15 


218-0 


1669 




8696 


59 


293-0 


476 i 




2S0i 


16 


216-4 


1673 


4 


8640 


60 


293-9 


468 




2448 


17 


219^6 


1487 




8784 


61 


294-8 


461 




25^3 


18 


222-6 


1411 




8928 


62 


296-9 


454 




2736 


19 


2255 


134S 




9072 


63 


297-0 


448 




2880 


20 


228-3 


1281 




9216 


64 


298-1 


441 




aos4 


21 


2310 


1225 




9360 


65 


299-1 


435 




8168 


22 


233-6 


1173 




9504 


m 


300-1 1 


429 




3312 


23 


2360 


1127 




9643 


67 


301 '2 


424 




34 Se 


24 


238-4 


1083 




9792 


^8 


302-2 


418 




seoo 


25 ; 


240-7 


1043 




9930 


69 


SOS-iJ 


412 




3744 


28 


243*0 


1010 




10080 


70 


- 304-2 


407 




3888 


27 


2461 


972 




10224 


71 


306-1 


402 




4m2 


28 


247-2 


940 




10363 


72 


306'1 


897 




417e 


29 


249-2 


911 




10612 


78 


307-1 


392 


2 


4320 


30 


251-2 


882 




10656 


74 


308-0 


386 




4484 


31 


253-1 


857 


5 


108(90 


75 


308-9 


382 1 




4608 


32 


255-0 


832 




10944 


76 


309-9 


878 




4752 


38 


256-8 


809 




11088 


77 


310-8 


373 




4898 


34 


258-6 


787 




11232 


78 


311-7 


369 




5040 


35 


260-3 


767 




11376 


79 


312-6 


366 




£184 


36 


262-0 


747 




11520 


80 


313-6 


361 




dms 


87 


263-7 


729 




11664 


81 


314-4 


357 




6472 


38 


2653 


711 




11808 


82 


315-2 


363 




5616 


39 


266-9 


694 




11952 


83 


816-1 


349 




5760 


40 


268-4 


678 




12096 


84 


816-9 


345 




5904 


41 


269-9 


663 




12240 


86 


317-8 


841 




6048 


42 


271-4 


649 




12384 


86 


318-6 


338 




6192 


43 


272-0 


636 




12628 


87 


319-4 


334 




6336 


44 


2743 


622 




12672 


88 


320-3 


331 


3 


6480 


46 


276-7 


609 




12816 


89 


321-1 


327 




6624 


46 


277-1 


697 


6 


12980 


90 


321-9 


324 




6768 


47 


278-4 


635 




13104 


91 


3227 


321 




6912 


48 


279-7 


574 




18243 


92 


3236 


318 



812 



THE 8TSAM ENGINE. 



[part 11, 



1 


II 


11 

11 


111 


iHill 


11 




1 
II 

e 

II 
no 


^1 

hi 


271 


13392 


93 


3:24*3 


315 1 




15S40 


3307 




13536 


H 


325 


312 1 




16560 


lis 


340-1 


2m 




130J>0 


95 


325-8 


3oa 


@ 


17'iaO 


120 


343'S 


250 




13924 


96 


32fltf 


30(J 




ISOOO 


125 


346-4 


241 




1396S 


^7 


327-3 


303 




18720 


130 


3496 


233 




HI12 


38 


328-1 


mo 


» 


10440 


135 


35:2-4 


225 




lit25fi 


m 


328-8 


297 




201S0 


140 


355-3 


217 




144 im 


urn 


32P-6 


2i^S 




20830 


145 


35S1 


211 


7 


16120 


1(15 


333-2 


28-2 


10 

1 


21600 


150 


360*8 


204 



GBNEBAL DESCRIPTION OF THE MODE OP ACTION OP THE STEAM 

ENGINE. 

Before proceeding to develope the general theory of the steam 
engine, or to investigate the mode of action of the steam in the 
cylinder, it will be well to enter into a general description of its 
construction and mode of action. In this preliminary description we 
shall only notice two forms of engine, namely, the non -condensing 
expansive engine, and the condensing engine without expansion. 

Plate VII. exhibits a sectional elevation of a condensing steam- 
engine of eight horse power, with its boiler attached. The steam 
being generated in the boiler A, is conveyed by the steam-pipe C 
into the valve- box or chamber I, which is shown upon a larger scale 
in fig. 1, plate IX.; from this chamber there are three passages, 
one (a) communicating with the top of the cylinder, the other (b) 
with the bottom of the cylinder, and the third (c) with a yeasel to be 
afterwards described, called the condenser. These passages are 
covered by a sliding valve, </, so formed that when in the position 
shown in fig. 1, all three of the passages are closed, but if it be 
moved downwards, and brought into the position shown in fig. 2, 
the upper passage a is made to communicate with the steam cham- 
ber I, while b and c are made to communicate with each other ; but 
if it be moved upwards, and brought into the position shown in fig. 
3, then the lower passage c communicates with I, and a and b with 
each other. The action of this valve being understood, if we snppose 
it to be in the position shown in fig. 2, we must immediately per- 
ceive that the steatn, which has been conveyed into the chamber I 
by the pipe C, will pass by means of the passage a into the top of the 
steam cylinder D, and by its pressure on the piston E give motion to 
it, and cause it to descend. This piston is connected by a rod e with 
a beam capable of turning about the center F, and the other end of 
which is connected by the rod ^ with a crank hy upon the shaft of 
which is fixed the fly-wheel O. It is therefore evident, ihftt any 



CHAP. VII.] THE STEAM ENGINE. 313 

motion of the piston £ will by means of the beam and connecting 
rods be immediately communiatcd to the crank shaft h, and cause it 
to revolve. Upon this shaft a contrivance termed an excentric (H) 
is fixed ; this is nothing more than a circular disc, which has the shaft 
or axis passing on one side of its center, the effect of which arrange- 
ment is to cause any point in its circumference to move nearer to or 
further from its center as it is made to revolve, by which means a 
reciprocating motion is given to the connecting rod k, which commu- 
nicates through the beam /, and rods m and n, to the slide valve d 
already described. Now, when the piston E, by the continued pres- 
sure of the steam upon its upper surface, has been forced to the bot- 
tom of the cylinder D, this valve d will, by means of the excentric, 
have been moved upwards, and brought into the position shown in 
figure 1, in which the communication between the boiler and the 
cylinder is closed, so that no more steam can be admitted to press 
upon the piston £. The downward motion, however, of the piston 
liaving been communicated, in the manner already described, to the 
fly.wheel O, the momentum which it has thereby acquired causes it 
to continue its motion, and as it carries round the crank ^, and the 
excentric H will produce the following twofold effect ; first, it will, 
through the instrumentality of the beam F, and connecting rods, e and^, 
reverse the motion of the piston, and cause it to commence its ascent 
in the cylinder; secondly, it will, by means of the excentric and 
system of rods attached to it, cause the valve d to slide upwards, to- 
wards the position shown in fig. 3. As soon as this motion of the 
slide valve commences, t)ie communication between the passage c and 
the box I being opened, the steam will now pass to the bottom of the 
cylinder, and there pressing on the lower side of the piston, will 
cause its reascent. 

We must now, however, notice what becomes of the steam which 
already occupies the upper portion of the cylinder. By reference to 
fig. 3, (which it must be recollected represents the position in 
which the slide valve is now supposed to stand,) the upper part of 
the cylinder will be seen to be in immediate communication, by 
means of the passages a and b^ and the pipe O, with the condenser 
M, which is a cylindrical vessel entirely surrounded with cold watet, 
and which also has a small jet of cold water constantly playing into 
iL If we now suppose the whole of the air to have been previously 
expelled from the vessel M, (the manner of doing which will be pre- 
sently described,) so that nearly a perfect vacuum is formed within it, 
it will readily be understood that the moment the communication is 
opened between it and the upper part of the cylinder, the steam con- 
tained in the latter will rush into this vacuum with a very great velo- 
city, and being there immediately condensed Jby the cold water 
playing into and also surrounding the vessel M, the vacuum will be 
preserved, and thus no resistance will be offered to the motion of the 
piston by the pressure of the steam upon its upper surface. As, 
however, the injected water, together with that resulting from the 
condeDBed steam, would in time fill the vessel M, an air pump, L, is 



314 THE 8TBAM BNOINB. [PABT II. 

made to communicate with it, which being worked by a rod from the 
beam of the engine, always keeps the condenser empty. And fur- 
ther, as the heat lost by the steam would in a short time so far 
elevate the temperature of the water surrounding the condenser as 
to render it incapable of continuing properly to condense it, it be- 
comes necessary continually to change the water, with which object 
a pump, P, worked by a rod from the engine beam, raises cold water 
from a well or other source^ and pours a continued stream into the 
vessel surrounding the condenser, while an equal quantity of the 
warmer water is allowed to run off. 

Before the engine is set to work, the cylinder D, the condenser 
M, and the passages between them, are filled with common air, 
which it is necessary to extract To effect this, by opening the 
valves a communication is made between the steam-pipe C, the space 
below the piston in the cylinder D, the eduction-pipe O, and the 
condenser M. The steam will not at first enter the cylinder D, or 
will only enter it a little way, because it is resisted by the air ; but 
the air in the eduction-pipe O, and the condenser M, is forcibly 
driven before it, and this part of the air makes its exit through the 
valve N, called the suift valve, and which is kept covered with water. 
The steam-admission valve is now closed, and the steam already 
admitted is converted into water, in the manner already described, by 
the coldness of the condenser M, and by the jet of cold water which 
enters it from the well SS, in which the condenser is immersed. 
When this steam is condensed, all the space it occupied would be a 
vacuum, did not the air in the cylinder D expand, and fill all the 
space that the original quantity of it filled ; but by the repetition of 
the means for extracting a part of the air, the remainder is blown 
out, and the cylinder becomes filled with steam alone. 

In order that the connecting rod e may work freely, and yet 
possess the desirable property of being steam-tight, it passes through 
what is called a stuffing or packing box. This stuffing consists of 
some material which the steam will rather adapt to its office than 
injure; leather, which answers well for the stuffing or collars of 
machines never to be subjected to heat, will not answer here; 
hempen yarn is the material usually employed. The rod of the 
piston / passes through a stuffing box of the same kind as that of the 
piston £ ; and the pistons themselves are surrounded with stuffing. 

The cylinder D is surrounded by a case, to keep it from being 
cooled by contact with the external air. The extremity, or any 
given point removed from the center of the great beam, can describe 
only the arc of a circle; but it is necessary that the piston rod e 
should rise and fall vertically. An apparatus is therefore used, 
called the parallel joint, which is easily understood by inspection. 
By this means the rod e not only rises and falls perpendicularly, bat 
is perfectly rigid, and communicates all its motion to the great beam 
in each direction of its motion. The connectiivj rod g does not 
require the same contrivance, because it does not rise and fall per- 
pendicularly; its lower end, with the outer end of the crank, describ- 



CHAP. VII.] THB STEAM ENGINE. 315 

ing a circle : it has therefore only a simple joint, admitting of this 
deviation. 

In order to communicate a rotatory motion to the fly-wheel, in- 
stead of the crank may he used a contrivance giving twice the rapidity 
to the fly. For this purpose, on the outside of the axis of the fly, 
irhere the crank is shown in the plate, a small toothed wheel is 
fixed, and can only he moved with the fly : at the extremity of the 
rod g^ and on that side of it which is next the fly-wheel, another 
toothed wheel is fixed, in such a manner that it cannot turn round 
on its axis, hut must rise and fall with the rod to which it is attached. 
These two wheels work in each other, and that attached to the connect- 
ing rod cannot leave its fellow, because their centers are connected by 
a strap or bar of iron. When, therefore, the connecting rod rises, the 
wheel upon it moves round the circumference of the wheel upon the 
axis of the fly. By this means the fly makes an entire revolution for 
every stroke of the piston, and some mechanics are apt to think that 
they are great gainers by such an arrangement : the contrivance is 
certainly el^ant, but with respect to utility, the fact is, that a crank 
is preferable ; for it is more simple, cheaper, and less likely to be out 
of order, while, if the fly be large enough to receive, with less velo- 
city, all the momentum that can be communicated to it, the efiect 
will certainly not be inferior. 

We now pass on to describe the non-condensing expansive engine 
shown in plate VIII., in which fig. 1 is a side elevation ; fig. 2 an end 
eleTation; fig.daplan; and fig. 4 plate IX., asection of the cylinder, show- 
ing the steam passages and valves. By a glance at this plate, the means 
by which the motion of the piston is transmitted to the crank shaft 
"mil be seen to be similar to that just described ; the real diflerence 
between them consists in there being no condenser and air pump in 
the engine which we are now describing, and in its having a second 
ezcentric by which the admission of steam to the chamber I from the 
boiler can be cut off at any desired portion of the stroke. The mode 
of admitting the steam to the top and bottom of the cylinder, alter- 
nately, by means of the slide valve d^ is precisely similar to that 
already described; but after the steam has done its duty in the 
cylinder, instead of passing into the condenser it escapes by the 
passage d, directly into the atmosphere. We have yet, however, to 
explain the use of the second cxcentric, and to show in what manner 
the steam is caused to expand in the cylinder, by which a saving is 
eflTected in the quantity of steam used by the engine. In the engine 
which we have just described, the steam is supposed to remain on 
during nearly the whole stroke of the engine, and to have therefore 
nearly the same pressure throughout the stroke ; but in that which 
we are now describing, the communication between the boiler and the 
cylinder can be closed at any period of the stroke that is desired, so 
that after the steam has been cut off, it then expands in the cylinder 
as the piston moves before it, and therefore presses upon the piston 
at each instant with a pressure due to its specific volume at that 
instant. 

The method of adjusting the exccntric rod, so as to cut off the 



316 THE STEAM BN6INE. [pABT II. 

Steam from the cylinder at any desired period of the stroke, is shown 
in fig. 5, plate IX., and is as follows: — ah is a bell-crank lever, 
by means of which the motion of the excentric is communicated to 
the valve, the arc cd being the distance (termed its effective stroke) 
through which the valve has to be moved to cut off the steam entirely 
from the cylinder; then, if we make the angle gkk equal to the 
angle ckd^ and taking the diameter of the excentric's path, or double 
the distance between the center of the excentric and the center of 
the shaft in the compasses, apply them along the two dotted lines, 
the point marked 6 where they fall upon both, or where the angular 
distance is equal to the diameter of the excentric's path, will be the 

f)oint at which the excentric rod must be fixed to the arm h of the 
ever, in order that the steam may remain on during the entire 
stroke. Then, fig. 6, representing the excentric's path, if it be 
desired to cut off the steam at any portion of the stroke, as at ^, i, 
or J, we have only to measure such a proportion of the semi- 
circumference of the excentric's path, and taking the length of its 
chord in the compasses, apply them in the manner already described, 
to the angle gkh^ and the distances from k^ at which they severally 
fall upon both lines, will be the distances at which the excentric rod 
should be fixed when it is desired to cut the steam off from the 
cylinder at any such portions of the stroke. 

Or, if we put / for the length of the arm a of the lever, f for the 
effective stroke of the valve or the distance erf, ^ for the diameter of 
the excentric's path, d for the distance from k to the point at which 
the excentric rod should be attached to the arm h of the lever, and n 
the fraction of the stroke at which it is desired to cut the steam off 
from the cylinder, then 

/^8in(90«) 

E 

For example, if / equal 6 inches, t equal 1*5 inch, and 1 equal 
3 inches, and we give to n the successive values *25, '5, "75, or 
suppose the steam to be cut off at J , \ and \ of the stroke, we shall 
have for the corresponding values of rf, 

, , C X 3 X -383 ^ ^ . , 

at J stroke d = z = **^ inches, 

1*5 

at ^ stroke d = = 8*5 „ 

1*5 

1 . , , ^ 6 X 3 X -924 

and at } stroke d = = 1 1-1 „ 

1*5 

THKORV OF THE STEAM ENGINE. 

Before entering uj)on the general theory of the steam engine, it is 
but right to mention tliat the manner in which we purpose consider- 
ing the subject is tjimilnr to that first laid down by the Comte De 
Pambour, in his very able work * upon this subject ; and at the 

* The Theory of the Steam Engine, by Comte De Pambour. PubUalaed bj 
John M'eale, 1839. 



CHAP. VII.] THB 8TBAM ENGINE. 317 

same time to acknowledge the great services which he has rendered 
b? his investigations, which have led to the adoption of a theory 
alike applicable to every form of engine, and working under all cir- 
cumstances. The propositions upon which his theory is founded, 
are thus stated by him at page 25 of his work :^— 

^^ From what has been stated, it plainly appears that we ground 
all our theory on these two incontestable facts: 1st, that the engine 
having attained uniform motion, there is necessarily equilibrium 
between the power and the resistance; that is, between the pressure 
of the steam in the cylinder^ and the resistance against the piston, 
which furnishes the first relation *, 

Pj = R. 

And 2dly, that there is also a necessary equality between the 
production of the steam and its expenditure, which furnishes the 
second relation t, 

_ w8 P 

And these two equations suffice for the solution of all the 
problems." 

The manner in which Tredgold, and other writers upon this sub- 
ject, had determined the power of an engine, was by measuring the 
area of its piston, and then, assuming that the engine would move at 
a given velocity, and that the pressure in the cylinder would be 
identical with that in the boiler, they thought that the continual 
product of these quantities would give them the load which the 
engine ought to raise in a given time, and to a given height, pro- 
Tiding no loss had arisen from the friction of the machine, and other 
canses; and further, supposing that the portion thus lost always 
bore the same constant ratio to the whole, in the same cla.ss of 
engines, they determined arbitrary coefficients, by which they multi- 
plied the above product, and they conceived that the quantity thus 
obtained represented the actual effective power of the engine with 
sufficient accuracy for all praqtical purposes, although no notice 
ivhaterer was taken in the calculation of the evaporative power of 
the boiler. 

A little consideration will, however, show the inaccuracy of this 
method of calculation. The engine itself is not the moving power 
by which we produce the desired effects; the boiler is the part in 
^vhich the power resides, and the engine is only the instrument or 
machine through which this power is transmitted, and by which it 
can be applied precisely at the point, in the mode, and with the 

* In whidi p, represents the pressure of the steam against the piston, per 
unit of surfaoe, and a represents the resistance of the load against the piston, 
divided in like manner per unit of surface. 

•f In which v is the velocity oi the piston under the resistance a, m the 
specific Tohime of the steam in the hoiler under the pressure p, or the ratio of 
Its Tohime to that of the water which produced ii^ s the volume of water 
er apo rafd by the boiler in a unit of time, and a the area of the cylinder. 



318 THE STEAM BNOINB. [paR*^ 

velocity which we desire. A steam engine may, in this respect^ 
compared with a crane, by means of which a number of met^ 
enabled to raise a considerable weight ; bat here no power resid.ef 
the crane, it is merely the instrument through which the mcT^ i 
enabled to apply simultaneously their united energies, and to pro^i 
certain effects; but those men could produce an equal amon^int 
effect without the crane, although not with so much ease or con ^ei 
ence. And, in like manner, the power which can be transmitt^ 
the engine can only equal that which resides in the boiler. For, 
in the crane, the power exerted through it must depend on tl 
number of men by whom it is moved, so also in the steam engfiK 
must the power which it can exert, or the amount of work which i 
can perform, depend only upon the power of the boiler. iVot 
however, that we mean to assert, that any amount of power which 
it is possible to derive from a boiler could be transmitted and app/fecf 
by the engine, for, as the crane has been designed and proportfooed 
only for the production of a certain effect, and the application of a 
greater power than that required for its production might strain and 
injure the machine, so it is with the steam engine, every engine is 
constructed only to transmit safely a certain amount of power frotn 
the boiler to the working point, and the attempt to produce a grease 
effect might be attended by the derangement of the machinery. 

If a equal the area of the piston in square feet, and p^ ^ 
pressure of the steam in the cylinder, per square foot of surface, tl^ 
product, or a/?, will obviously represent the resistance which ^ 
engine is capable of overcoming; and if this be multiplied by ^ 
length of the piston's stroke = /j , we shall have the whole amo^ 
of work which the engine will perform in each stroke, or represent^ 
this amount by k'i , we shall have 

», = a/,/?, (X.) 

This equation, however, supposes the steam to act with C 
pressure pi during the whole stroke of the engine, and therefore on 
applies to engines in which the steam is not cut off until the end • 
the stroke, and which therefore do not work expansively. 

In order then to obtain a general expression which will indac 
the expansive engine also, let us suppose that /j represents only th 
portion of the stroke during which the steam is not cut off, in whi< 
case the above formula will still represent the work done by tl 
engine during such portion of the stroke. Now, if x equals tl 
length of a portion of the cylinder, equal in content to the stei 
passages and the space left at each end of the cylinder for the clei 
ance of the piston, we shall have a (/, + >^) for die actual space occ 
pied by the steam in the cylinder (having a pressure of />, ) at t 
moment when the steam is cut off; then, if/ represent the length of t 
stroke from its commencement, at any moment after the expansi 
of the steam has commenced, and p the pressure of the steam d 
to its altered volume, we shall have from formula (IX.) 



/, + X /w \ n 



CHAP. VII.] THB STEAM ENGINE. 319 

Then, since the work done by the steam in the cylinder is equal 
to the pressare upon the piston multiplied by its area and the length 
of the stroke, or the distance that the piston has moved under that 
pressare, if A / represent the distance moved over by the piston after 
the steam has been cut ofif^ we shall have for the work performed 
daring such motion, 

a/7 a/, 

in which, substituting for p its value derived above, we have 

therefore, the whole amount of work done by the steam during its 
expansion in the cylinder will be represented by 

or putting w, for this amount of work, and /, for the total length of 
the stroke from its commencement, 

«nd int^rating between the limits of /, and /,, we have 

». = a (/. + X) hyper. Wj^ (^ + ^ ») " f « (^' - ^•) - (^I) 

Now, if we add together the work done before expansion, (= t0„) 
and the work done after expansion, ( = «;„) and represent the whole 
work performed by the engine each stroke by W, we shall have 

W = w, + w,; 

in which, substituting for w^ and w, their values derived from equa- 
tions (X.) and (XI.), and reducing, we obtain 

W=a Q + ;».) [K + (/. + A)hyper. log \±^^ - ^ a /.....(XI/) 

Then, if P represent the pressure or resistance which the work 
W exerts against every square foot of the surface of the piston, 
W = Pa^, and therefore 

P/. = d + />.) {/. + (/. + X) hyper. logjf^} -]ir - (XII.) 
wbich ezpreanon becomes 

P=J», (XIII.) 

when l^=zli^ or when there is no expansion. 

These forrouls, then, enable us to determine the effects which 
would be produced by any engine working under given circumstances; 
hat it b^mes a matter of considerable practical importance to 



320 THE STEAM BN6INB. [PART II. 

determine those values of the several quantities which shall produce 
the greatest mechanical effect with the least expenditure of steam ; 
or, in other words, the quantity of water evaporated by the boiler 
remaining constant, shall give to P its maximum value. In the high- 
pressure engine, working without expansion, this takes place when 
the pressure in the cylinder is as nearly as possible equal to that in 
the boiler. In the expansive engine, the same law holds with regard 
to the pressure at which the steam should be admitted to the 
cylinder to produce the greatest mechanical effect with any given 
expansion. But there is in every engine working by expansion a 
certain proportion of the stroke through which the expansion will 
produce a greater effect than through any other proportion, and this 
will be easily found after we have obtained expressions for Q, or the 
number of cubic feet of water which the boiler is capable of evapo- 
rating per minute. 

If V. equals the specific volume of the steam at the pressure /?, , 
at which it is admitted to the cylinder, we shall have for the num- 
ber of cubic feet of steam of that pressure, generated per minute, 
Q f 1 ; and if the engine makes » strokes per minute, each equal 
/j feet in length, »a(/j + ^) will equal the number of cubic feet 
of steam at the pressure p^ used by the engine per minute, and 
therefore from the second principle laid doivn by De Pambour*, 
we have 

Qr, = ar(/, H- >); 

then, if V equal the velocity of the piston in feet per minute, 

V 
we have V = i-/^ , or » = - ; and also from equation (VIII.) we 

have V, = ; therefore, substitutine these values above, 

we obtain 

Q aV(/, +x) 



n + qp^ I, 

whence we obtain for the value ofp j , 
_ l,Q n 



(XIV.) 



^* ayV(/, +A) g 

We have also seen by equation (XIII.) that in an unexpansive 
engine /^j =: p, p being the resistance against each square foot of 
tlie piston's surface, arising from its load, from the friction of the 
engine, and from the pressure of the atmosphere, or imperfectly 
condensed steam on the other side of the piston. Now, taking the 
most simple case, or that of the stationary engine, if we put ^ for the 
resistance occasioned by the load upon each square foot of the 
piston's surface, J* ^^'^ ^^^ resistance arising from the friction of the 
engine when unloaded, and ^ ^ for the additional friction produced 

* See page 317. 



CHAP. YII.] THB STB AM BNOINB. 321 

by the load ^ upon eyery square foot of the piston s surface, and also 
r to represent die pressure of the atmosphere or uncondensed steam 
upon each square foot, we have 

and substituting this value of jp^ in equation (XIV.) above, 

";*»>+/^'=.-?vTi7T^-? <^^> 

wbich equation expresses the relation between all the several quan- 
tities for unesfpansive eneines. 

In exfMUuive engines 7^ represents only that portion of the stroke 
which IS performed before the steam is cut off, the whole stroke 

being equal to /^ ; therefore in this case ' = j- > cuid equation 

(XIV.) becomes 

'.=j?4rnr)-F <''^'' 

Now, from equation (XII.) we have for expansive engines, 

/, +(/,+x) hyper, log ^-^ ^^ 

•nd substituting for P its value determined above, reducing, and 
solving in respect of ^, we have 

Q(rV, + typer.log^j) /+. + ^ 
( = ^1 ^ ^i, I M+^-^ - _ 1 ... (XVII.) 

Then, aboe the actual useful mechanical effect produced by the 
engine per minute equals ^ a V, we have for the power of the engine 

^ ^(rTi + ''yp-'^°grT-D «v(/-H^4-f) 

«"^ — TUTf^ — 1 + * 

Now, in order to arrive at that value of /^ which will give the 
maadmom value to ^ aV, or, in other words, to ascertain at what 
|iortion of the stroke the steam should be cut off in order to produce 
the greatest effect, let us substitute for a V its value derived from 
eqvfttion (XIV.), whence by reduction we obtain 

9(1 +f)V/, + X^ '*^ */, +x l^+?, n+qp^ J 



322 THE STEAM ENGINE. [PART II. 

then, differentiating in respect of /^ and remembering that since the 
above expression is a maximum its differential will = 0, we have 






whence, reducing and solving in respect of j , we obtain 



(XVIII.) 



equal that portion of the stroke (the whole stroke being unity) at 
which the steam should be cut off to produce the greatest effect, with 
the least expenditure of steam. 

DESCRIPTION OP THE VARIOUS KINDS OP ENGINES, AND THE FORMULAS 
POR CALCULATING THEIR POWER. 

Although almost every manufacturer of steam engines adopts his 
own peculiar form of construction, by which a great diversity in 
their external appearance is occasioned, the principles upon which 
they act are similar in all engines of the same class; and, therefore, in 
attempting a classification of the different kinds of engines, we shall 
pay no attention to the details of their construction, but only regard 
the mode of employing the steam in the cylinders for the purpose of 
producing any particular species of motion. Proceeding, then, upon 
these principles, the following table represents at one view the 
various forms under which steam is at present employed as a 
moving power. 



Engines working | Non-condensing [ f^^'^, 
«m^/ expansion | Condensing. ^ 



Engines working 
with expansion 



r"-"- [c^^ j-^-«- 

^ J Steam pressing only on the 

c;,i»io i.^:»» J t upper surface of the piston. 
V Single acting j | st^^„,^i„^ ^^U on the 



I Steam pressing only on the 
lower surface of the piston. 



Stationary non-eandenBtng EngineB without expansion. 

The mode of action of the steam in the cylinders of this class <- ^ " 
engines is precisely similar to that already described at page 31^^ 
and a view of which is given in Plate VIII., with the excepUon tbi^^ 
the steam is never cut off from the valve-box I, and therefore pr ess e d 
with an equal pressure upon the piston daring nearly the who^ff 
extent of the stroke. With regard to the mechanical coDatmction a^ 
this kind of engine, it depends in a great measure upon the fancy c^- 
the maker, or upon the purposes to which the engine is to be applied ^ 
there is perhaps a greater diversity in the form of this class (^'« 
engine than of any other. For particular examples of the method 



CHAP. VII.] THE STEAM ENGINE. 323 

of coDStnictiDg all kinds of CDgines, we must refer to the last edition 
of Tredgold's elaborate work upon the steam engine, which, with 
its appendices, forms one of the most magnificent works ever 
published. 

The formulse for calculating the power of engines of this class 
may be immediately derived from equation (XV.) already given, by 
solving in respect of any of the quantities which are wanted. Thus, 
we obtain for the values of V^ Q, and ^, 

v= ^^. 



or, multiplying ^ by Vo, which then gives the actual work performed 
by the engine in each minute, we have 

We may reduce these formulae to a more practical form by insert- 
ing the numerical values of those quantities which are either constant, 
or sufficiently so for all practical purposes. Thus, t being in this 
case the pressure in pounds of the atmosphere upon each square foot 
of the piston's surface, equals 2118 lbs. ; for n and ^, we have the 
values already given for non-condensing engines at page 310, and 
assuming that x = *05 /„, ^ = '14, and y = 144, as adopted by 
De Pambour, the above formulae become 

^ "■ a(6'95536 + 0027531 f)' 

Vg (6-95536 -f '0027531 e) 
^ ■" 10000 

^aV = 3632268 Q — 2526-37 a V. 

Locomotive Engine working without expansion. 

The mode of action of the steam in the cylinder of a locomotive 
engine, does not in any way differ from that in the engine already 
described; the only essential differences of which we have to take 
notice being in the details of its construction, and in the nature of the 
work which it has to perform. 

Plate X. represents the longitudinal, and Plate XI. two transverse 
sections of one of the engines employed on the Great Western Rail- 
way. The principal peculiarity in the construction of a locomotive 
consists in the form of the boiler, which, in order to lessen the weight 
fo be moTed, is much smaller than the ordinary boiler for other 

Y 2 



324 THB 8TBAM BNOINB. [PABT II. 

engines of the same power, and wbicb, therefore, requires a very 
intense heat in the furnace, and a large evaporating surface: the 
first of these is effected by emp]o3ring coke instead of coal, and pro- 
ducing rapid combustion by a strong draught; and the second bv 
causing the water entirely to surround the fire-box, or furnace, and 
by carrying the smoke and heated air from the same through a 
number of small tubes which are likewise surrounded witli water. 
Thus, it will be seen by an inspection of Plate X., and also of fig. 2, 
Plate XL, (which latter is a transverse section through tlie fire-box,) 
that the water not only covers its upper surface, but that it surrounds 
it on every side, leaving only a small space on one side for the 
opening, I, by which the stoker regulates the fire. The tubes aa^ 
to which we have already alluded, for convejang the smoke from the 
fire to the chimney, are ninety-one in number, and expose a very 
large heating surface to the water ; and in addition there is a hollow 
bridge, £, also filled with water passing through the center of the 
fire. The bars, X X, of the furnace, are so arranged, that when it 
is desired to put out the fire they can all be simultaneously lowered, 
allowing the burning contents of the furnace to fall upon the ground, 
or into a vessel placed to receive them underneath the engine. The 
sides of the fire-box, being flat, would be liable to be forced out by 
the pressure of the water and steam, to obviate which the sides are 
held together by short rivets, and the top is strengthened by a number 
of short girders ^, bolted to its upper side. The flat ends of the 
boiler are prevented from being forced out by the fire tabes aay 
which are secured at each end, and by tie rods which run from one 
end to the other; the other parts of the boiler derive snflicient 
strength from its cylindrical form. There are two safety-valves, one d, 
under the immediate inspection of the engine-driver, and which by 
means of a spring manometer Q, always indicates to him the pressure 
existing in the boiler ; the other L, is placed (as an additional pre- 
caution) beyond his reach, so that it cannot be meddled with, or have 
its pressure altered. A glass tube W, fig. 1, Plate XL, made to 
communicate with the boiler at both its upper and lower extremities, 
so that the water in it always stands at the same level as that in the 
boiler, serves to indicate to the engine-driver when the quantity of 
water falls short. An opening R, called a man-hole, is left in the 
upper part of the boiler, being covered with a screwed lid, in order 
to enable the interior of the boiler to be examined ; c is a whistle, 
so contrived, that the steam being allowed to rush forcibly through it— - 
produces a very shrill whistle, which is employed as a very generaL^ 
and useful means of signaling on railways. It will be obsenred that^ 
the boiler is surrounded by an exterior case, a small space being lefk:^ 
between and filled with air, which prevents in a considerable degress 
the loss of heat from radiation, that would otherwise take place. 

In consequence of the steam space in a locomotive boiler beings 
very limited, the steam is formed almost as it is used, and immedi-' 
atefy the communication with the cylinders is opened and the steaitt 
allowed to flow into them, a violent commotion in the water in thff 



CHAP. VII.] THE STEAlf ENGINE. 325 

.boiler is occasioned by tbe sudden production of steam to supply its 
place, the result of which is, that a large quantity of water, in a state 
of minute mechanical division, is carried over with the steam into the 
cvlinders, and there gradually accumulating, would at length materi- 
ally interfere with the working of the engine. This effect, which is 
called primingy is to a certain extent obviated by a dome and steam- 
chest (H) heing formed at the top of the boiler, and by the entrance 
of the steam pipe G by which the steam is conveyed to the cylinders 
being carried to the upper part of this chest, and having its mouth 
turned upwards, by which the space for steam is somewhat increased, 
and more time is given for the separation of the water from the 
steam. The admission of steam to the cylinders is regulated by the 
Talve F, which the engine-driver can open or close by means of the 
handle d. The steam pipe has an expansive joint Z, which allows of 
a slight elongation or contraction of the pipe, in consequence of its 
varvms temperature. The steam pipe passes straight through the 
boiler mto the chamber in the fore part of the locomotive, termed the 
smoke-box, it there divides as shown in fig. 1, Plate XL, into two 
hrancbesX/y ^ch leading to one of the cylinders; the manner in 
which the admission of the steam alternately to the top and bottom 
of the cylinder is effected by the slide valve T, is precisely similar to 
that alr^y described at page 312. But in a locomotive it is neces- 
sary that Uie engine-driver should possess the means of making the 
engine revolve in either direction, in order to effect which there are 
two excentrics to each slide valve, fixed upon the crank shaft, oppo- 
site to each other, and so arranged that, by means of a lever within 
reach of the engine-driver, he can connect either excentric with the 
slide valve, and by reversing its motion also reverse that of the 
engine itself: only one of these excentrics (C) is seen in the plate. 
The motion of the piston n, is transmitted to the crank »«, by the 
piston rod o, and connecting rod r; the piston rod being made to 
move straight by the cross head B, moving in guides. After the 
steam has performed its office in the cylinders it is expelled through 
the pipe M, into the lower part of the chimney, where by its partial 
condensation it produces a powerful current of air through the fire 
and tubes aa. The water formed in the cylinder by condensation, 
or that which is brought over by priming, is allowed to escape by the 
cock P. The supply of water to the boiler is maintained by a small 
force-pump D, the plunger of which is attached to the cross head B 
of the piston rod, and which draws tbe water from the tender 
through the pipe O, and forces it into the boiler. The cylinders, 
bein^ entirely within the smoke-box, are always maintained at a 
considerable temperature, by which means very little loss results from 
condensation of steam in the steam passages or cylinders. 

The nature of the resistances to which a locomotive engine is 
subjected are somewhat different to those of a stationary engine, 
arising from the two following causes, viz., the resistance occa- 
sioned against the piston, by the ejected steam being blo\^ii forcibly 
throDgb tbe contracted pipe M into the chimney, for the purpose of 



326 THB STEAM ENGINE. [PART II. 

producing a draught in the fire, as already described ; and the resistance 
which the air occasions to the progressive motion of the train. The 
former of these, the Comte de Pambour, by his experiments, has 
ascertained to increase directly as the velocity of the piston, and to 
amount upon an average to about 1*75 lbs. per square inch of the 
piston's surface, when moving with a velocity of 150 feet per minute, 
or putting tt ^ V for this pressure per square foot, at the velocity V, 
we have 

^1 : 1-75 X 144 : : V : 150; 

.-. «-j = 1-6848. 

The second resistance, both theory and experiments show to in- 
crease as the square of the velocity, and from the results of De 
Pambour's investigations it appears that for a train of average length, 
moving at a velocity of ten miles per hour, this resistance amounts to 
about 33 lbs., which, assuming the engine to be of the most usual 
dimensions, is equal to '0055136 lbs. for every square foot of the 
piston, when moving with a velocity of one foot per minute; or, 
putting r for this resistance, we have rV^ when the velocity is V. 
As the value of r depends not only on the number and description 
of the carriages composing the train, but also on the size of the 
driving wheels, the diameter of the piston, the length of its stroke, 
the gauge of the railway, and even on the direction and force of the 
wind, it will be perceived that the value which we have assumed 
above is very general indeed, and that, wherever great accuracy is 
required, these circumstances must be taken into consideration, and 
the more exact value of r deduced therefrom. 

We have therefore, in this case, for the value of P, or the sum of 
the resistances against the piston. 



V = 



m(X 
of V, 

l^ 

a(l, + ^){n + g[{^ + rV^)(l + (?) +/+ tt + », V]} ' 



which being substituted in equation (XV.) for ^ (1 + ^) -f /"+ «-, aDd_^ 
reducing, we have for the values of V, Q, and ^a Y, as follows : — 



Or, substituting for n, ^, x, v and ^, the same numerical values i^b^ 
before, and making /*= 3*0125 lbs. per square foot of the piston'^ ' 
surface, when moving with a velocity of one foot per minute, whic^-^ 
would be its average value, these formulae become, 

10000 Q 

^ "■ a (6-6143 + 002753^ + -00001518 7* + 00406887)' 



CHAP. VII.] TH£ STEAM ENGINE. 327 

Q = fQQ55(C*61*3 + -002753 e + -00001518 V« + 0040688 V); 
^aV = 3632400 Q — a (-005514 7^ -f 1 -4779 72 + 2402-49 V). 

Condensing Engines working withotU expansion. 

One fonn of this class of engine is shown in Plate VII., and its 
general principles and mode of action have already heen descrihed 
at page 312. This class includes the greater numher of engines 
employed for propelling steam-hoats, which, although different in 
form, are identical in principle with the' engine shown in Plate VII. 

The general formulae for calculating the values of V, Q and f a V, 
are precisely the same as those already given at page 323, for non- 
condensing engines, and therefore need not he recapitulated here; 
but the practical formulse differ in consequence of the coefficients 
and constants having different numerical values. The value of 7, 
or the resistance on the under-side of the piston, is in this case much 
smaller than in the preceding, and may be taken on the average as 
only equal to 576 lbs. on each square foot ; the value of n and q 
for condensing engines will be found at page 310, and those of x, 
^, and fy may be taken the same as tliose already given at paee 323, 
for stationary non-condensing engines. Substituting these values in 
the general formulae, and reducing, we obtain 

_ lOOOOQ 

^ "■ a (2 3943 + 0030883 e) ' 

Y g (2-3943 -f 6030883 {) 
10000 ' 

^ a V = 3237280 Q — 77528 1 a V. 

Stationary non-condensing Engines^ working ea!pansivefy. . 

This engine, which is shown in Plates VIII. and IX., has been 
already described at page 315, and it therefore only remains to 
deduce the formulae by which the values of V, Q and faV may 
be obtained. 

Now we have, at page 321, 
and from formula (Xyil.)> by reduction, we immediately obtain 



828 



THB 8TKAM BNOINB. 



ygfn-f y[g(l+<P)-hj 



[part II. 



^ + hyper, log ±± 



A 



And subsUtuting in these formulae the nnmerical values of the 
constant quantities, viz., w equal the pressure of the atmosphere, 
equal 2118 lbs., n and g as given for non-condensing engines, at 
page 810, and x and ^ as before; also, eiving to/* the value assigned 
to it by Comte de Pambour for this class of engine, viz., 864 lbs. 
per square foot of the piston's surface, we have 

eav = 3813883 QJr-4- + l»yP«'- log r-'"-! - 3157-73 a V; 



10004 Q 



V = 






aC8-2796 + -002622 f) 

yg (8-2796 -f -002622 e) 

In order to simplify the calculation of these Quantities, and ^^r 
the benefit of those who are not conversant with the use of !<ȣ- 

arithms, we subjoin a table of the values of the expression 



/. 



/i + x 



»i -r A 

hyper, log j , for values of j-, (or the fraction ezpressmg tYjat 

portion of the stroke which is performed before the steam is cut o^j 
varying from one-tenth to nine-tenths of the whole stroke. 



Value 


Value of 


Value 


Value of 


-t 


^ + hyper.log'« + ^ 


-'t 




•10 


2-61258 


•60 


166558 


•15 


2-40823 


•65 


1 •47628 


•20 


2-23507 


•60 


1^40366 


•26 


208610 


•66 


LS1218 


•80 


1-95576 


•70 


1^26981 


•33 


1-87721 


•76 


1-20943 


•85 


1 •84008 


•80 


1-16248 


•40 


1-78619 


•85 


109862 


•45 


1-64194 


•90 


104TS9 



Loeonwtive Engine working expaiuively. 
Lately the principle of working locomotive engines ezpannic 
has been successfully adopted, and we therefore give the fumf 



CHAP. VII.] THB STB AM BNOINB. 329 



for calcolating the effects which thej will then produce. 
In this case we have 

*"'^ 7UT7) 






^'--«rv'. 



And snhstitnting for the constant quantities the same numerical 
ralues as those already employed at page 326, for locomotires work- 
ing without expansion, we have 



^ IOOOOQ(.Ji_.^.Hhyper.log^^) 

a (6-2993 -f -002622 ^ + 00001446 V' + -003875 y) ' 
V a(62993 + 002622 e + '00001446 Y* + -003875 v) 
10000(^^^ + hyper.log^^) ' 

^a V = 3813882 Q ( jA_ + hyper. log ^^^ ) 

— a (-005514 y» + 1-4779 y« + 2402-49 V). 
The apparent complication of these formula will disappear when 

we snhstitute for f j— ^ — + hyper, log / ) its value from the 

\ *i + ^ *i + ^ / 

sbove tahle, which cannot he done until the proportion of the stroke 
at which the steam is cut off is known. 

StaHanaiy Condensing Engines^ with one Cylinder working 
eapaneively. 

This enffine only differs from that already described at page 312, 
in its wonung expansively, which is effected by the agency of a 
second excentric, in the manner explained at page 315. 

The general formula for the values of V, Q, and e^^i ^^^ identical 
with those already given at pages 327 and 328, for non-condensing 
working expansively; but the practioJ formuln differ in 



330 THE STBikM ENGINS. [PjkRT II. 

consequence of the constants t, w, y, ?., ^, and f^ having in this case 
the value assigned to them at page 327, for condensing engines working 
without expansion ; these heing substituted in the general formulee, 
they become as follows : — 

10000 ^ /"-^ - i^-™ i.«'« + ^ 



Q = 



a (20953 + 0029412^) 
Va (2*0953 -f '0029412 6) 



V = — 

a (20953 + 0029412^) 



^a V = 3399972 Q (j-^ -i- hyper, log -2-^) — 712-123aV. 

Condensing Engines, with two cylinders^ working expansively. 

This form of engine was invented by Mr. Arthur Woolfe, and 
patented by him in the year 1804. The peculiaritv in its mode of 
action consists in employing two cylinders, through both of which 
the same steam is made successively to pass. The details of its con- 
struction will be better understood by reference to Plate XII., in 
which fig. 1 is a section of the two cylinders and slide valves. 
Supposing the various parts to be in the positions shown in this 
figure, the steam from the boiler enters by the aperture A, and 
passes through the passage B into the top of the smaller cylinder 
C, and there, pressing on the upper surface of the piston, pro- 
duces a downward motion; when the piston has performed a 
certain portion of its stroke the communication with the boiler is 
closed, so that the steam acts upon the small piston only by its 
expansion during the remaining portion of its stroke. The rod a 
of the small piston, and that b of the large one, being both connected 
to the same beam, descend together, during which the steam which 
occupied the space below the small piston passes through the 
passages D and E into the space above the larger piston ; but, since 
the content of the cylinder F is larger than that of C, in doing so it 
expands and produces a pressure on the upper surface of the large 
piston, the under-side of which is in immediate communication with 
the condenser by means of the passage O and aperture H. As the 
pistons descend, the slide valves, moved by an ezcentric in the 
ordinary manner, change their position, and during the up-stroke of 
the pistons are in the situation shown in fig. 2, in which it will be 
seen that the steam from the boiler now presses directly on the 
upper side of the smaller piston, while the top of the small cylinder 
now communicates with the bottom of the larger one, by which the 
effort of the steam, in expanding from one cylinder to the other, is 
exerted on Uic under surface of the large piston, the upper side of 
the same being now in direct communication with the condenser by 
means of the passage £ and aperture I. The mode of action of the 



CHAP. VII.] THE STEAU ENGINE. 331 

steam in the cylinders being then correctly understood, it is only 
necessary to state, that the manner of transmitting the motion of the 
pistons to the working point, and the arrangement of the condenser 
and air pump, may be the same as in other steam engines, and 
similar to that which we haye already described. 

In order to deduce formulee for calculating the effects of this class 
of engine, let a^ represent the area of the small piston, and a^ that 
of the larger piston, both in square feet; let /, equal that portion of 
the small piston's stroke which is performed before the steam is cut 
off, l^ the whole length of the small piston's stroke, and l^ the 
length of the larger piston's stroke ; also, let p ^ be the pressure of 
the steam in the small cylinder before the expansion commences, 
Pa the pressure of the same in the large cylinder before expansion, 
and ^, and A^ the length of a portion of each cylinder, equivalent 
in content to the space left for clearance and the steam passages. 
Then we have from equation (XI.*), page 319, for the work down 
in the small cylinder, 

«i (^ -rp, ) 1^ -f (/x + Xi) hyper, log ^^^^ - ^«i U- 

Now, in order to obtain the work done by the expansion of the 
steam in the large cylinder, put «, = a, /^ = the space moved 
through by the small piston, «, =0^/3 = the space moved through 
by the large piston, and « ^ a, / = Vie space moved through by the 
same at any portion (/) of its stroke. Then, the space occupied by 
the steam (having a pressure equal to p^) before the commencement 
of the stroke, equals 9^ -f^i^i -f-^^Xg, and that which it occupies 
at any portion (/} of the strolce equals «-fa,X| -{• a^x^; therefore, 
patting |3 = ajAj H-a^Ag, we have from formula (IX.), for the 
preasare of the steam due to its altered volume, 






n \ 



Then, since A« represents the elementary space moved through by 
the larger piston, we have for the work developed during the motion, 
pAty which, by substituting the above value of/?, becomes 

^. As /n \ n 

therefore, the whole amount of work developed is represented by 

('.+^)J:i-^(- +/»,}-- J:a.; 

which being taken between the proper limits of « = «| , and « = «j| ' 
equals 

<"*«/:'.-Ti(=+'.)-5/:-^ 



332 THE 8TBAM BNOINB. [PABT II. 

whence, by integnting, we obtain 

(,, +^)hyper.log^l±| (J +/>t ) - ^(*, "'i)- 

Then, in order to eliminate />,, we can derive its value from 
equation (IX.) in terms ofp^ , viz., 

which being substituted for /?, in the expression above, it becomes 

and again, substituting for «^, x^, and 0, their several values above, 
and reducing, we have 

-x(^ +^)(- +/'.) hyper, log ^-l^^-^:^^!^^^ 

for the whole amount of the work done by the expansion of the 
steam in the large cylinder, minus the resistance produced by the 
reaction of the steam in the small cylinder ; whence, adding this to 
the expression already obtained for the work done in the small 
cylinder, and reducing, we obtain 

. -f hyper, log ^^f/-^^-J;-^"-^n)--^a,/3, 

which equals the actual work performed by Woolfe's engine, durin^^ "I 
one stroke. 

Then, in order to obtain the value of P, in this case the resistances^ 
against each square foot of the lar^r piston's surface, arising fron^^* 
the load and friction of the engine, and from the imperfect vacuun^^^ 
in the condenser ; let w, /, and ^, represent the same quantities a ^^^ 
before, only taken per unit of the large piston's surface; thei— *^ 
we have 

P = ^(l +^)+/+^, 

which being multiplied by a, /,, equals the whole amount of rcr " 
sistance developed during one stroke of the engine ; equal 

Therefore, since where the engine has attained uniform motion the 
resistance must equal the work performed, we have 



Q- 



CHAP. VII.] THB nUU XNGIMB. 833 

■h hyper, log ^^^^;^-^\>-^^^^n)-%,/3 

Then, if V equals the yelodty of the larger piston, and 9 the 

V 

nnmber of strokes which it makes per minute, we have » = -j- , 

*s 

whence we obtain, for the value of j^^, from equation (XIV.) 

* «iyV(/, +Xi) g' 

Then, snbsdtuting this valae of />, in the above expression, 
rednciug, and solving in respect of Q, V, and ; a, V, we have 

«.Vf» + <rr> (! + »)+/+>]? 
-ll- + hyper, log ^±-i. + hyp«. log •«J''Jt-*«>-+-"l^J 

Q {^ + Hyper. U^ ;4il + hypT. U«^f±^4±fJ} 

-«.v £. 

i + f 

In order to put these formulsB under a more practical form, we 
must snhstitute for the constants and coefficients their average values, 
vis., for n, 7, and ar,as before, for condensing engines, /*» 125 lbs., 
9 as before, x, = 'OS/j, and x, = '05/,, then we obtfun 

ie>«0Q{^24-H hyper. log;^^hyper.U.^4,^^±S^} 
^ " «, {2-231S + •0029412 {) 

(•,V-S9D007SQ {•9624 + hyper, log ^±^' 



334 THB STBAM RNOINB. [PART II. 

The following table, giving the numerical value of the expression 
hyper, log i±^+h.yper.logji^±^\±^) for jslues of 

-p varying from 77; to - , and for the three cases where — eqaals 

4, 3 and 2, may sometimes be found useful : in this table the two 
strokes are assumed to be of the same length, that is, /^ == Z^. . 



( 



Value of 


Value of (hyper, log ^ + hyper. lo| 


«s((i + ^)+M,\ 


^«i(/, + ^.) + W 








when a, « 4a). 


when a, « 3a|. 


when a, = 2a,. 


•10 


316969 


2-92674 


2-57162 


•15 


2^88200 


2-63906 


2*28393 


•20 


2-65886 


2-41591 


206079 


•25 


2-47654 


2-23359 


1-87847 


•30 


2-32239 


207944 


1 72432 


33 


2-23901 


1-99606 


1 64094 


35 


2-18885 


1-94591 


159078 


•40 


2-07107 


1-82813 


1-47300 


•45 


1-96561 


1-72277 


1-36764 


•50 


1-87088 


1-62793 


1-27281 



Sin^e-acting Engine^ in which the Steam cuUa only upon the upper 
surface of the Piston. 

This class of steam engine is that usually known by the appella- 
tion of the Cornish pumping engine, for which purpose it is more 
peculiarly adapted ; and from the surprising amount of work which 
has been realized by means of it, with the consumption of a given 
weight of fuel, has attracted a great deal of attention, and has become 
almost universally adopted in mining districts. 

As it is not our province here to enter into a detail of tbe 
mechanical construction of these engines, we have only shown in 
plate XII, figure 3, a section of the cylinder and valves ; it maj, 
however, be remarked, that these engines exhibit the greatest me- 
chanical skill in their design and construction, and doubtless ooe 
cause of the high duty which they perform results from the accuiacj 
and precision with which the various parts are formed, and the 
precautions adopted for preventing loss of heat by the steam io 
passing through the engine. 

In the plate, A is the cvlinder, which is surrounded by an outer 
casing sufficiently large to leave a space of about an inch round the 
whole of the exterior of the cylinder, which space (termed the steam 
jacket) is always kept filled with steam of the same pressure as thtt 
m the boiler, by means of the pipe B, which is in direct communici- 



CHAP. YII.] THB STEAM ENOINB. 335 

tioD with the hoiler ; a similar space is also left helow the cylinder, 
and kept constantly filled with steam. The ohject of thus sur- 
rounding the cylinder with hot steam, is to prevent its interior 
surface from heing cooled during the up-stroke; and for the same 
purpose, the top of the cylinder, as also the upper part of the valve 
box, and the entire steam pipe from the boiler to the cylinder, are 
enveloped in an external case, the space between being filled with 
sawdust or some other non-conducting substance. 

The piston being at the top of the cylinder, the valve C is opened^ 
allowing the steam to pass from the boiler into the cylinder, where, 
pressing on the upper surface of the piston, it causes its descent, 
raising in so doing the pump rods and counterweight which are 
attached to the other end of the engine beam. After the engine has 
performed a portion of its stroke (varying from -^^ to ^ in 
different engines), the valve C is closed, and the remainder of the 
downward, or, as it is termed in Cornwall, the in-door stroke, is 
performed by the expansion of the steam : the under -side of the 
piston is in communication with the condenser by means of the pipe 
D, which is closed by the valve E towards the conclusion of the 
down-stroke. There being in the Cornish engine no crank, or other 
means by which the precise length of the stroke is determined, it 
becomes necessary to adopt some means of preventing the piston 
descending too far, and by doing so injuring the bottom of the 
cylinder ; the way in which this is effected, is by so regulating the 
quantity of steam admitted through the valve C at the commencement 
of the stroke, that it shall be just sufficient to carry the piston to the 
proper distance, and no further; but since any variation in the 
pressure of the steam in the boiler, or in ihe quantity of water 
raised by the pump, would immediately destroy this adjustment, and 
cause the piston either to fall short of or to exceed its proper stroke, 
springs are placed so as to receive and stop the beam of the engine 
when it descends too low, and at the same time a bell is rung, which 
warns the engine-driver to lessen the quantity of steam admitted by 
the valve C. The piston having been brought to a state of rest at 
the bottom of the cylinder, the valve F is closed bv which the com- 
munication with the condenser is cut off, and the valve G, termed the 
equUihrium vaive^ is opened, by which a free communication is effected 
let ween the top and bottom of the cylinder, and an equilibrium esta- 
blished between the pressure on the upper and under side of the piston, 
which, therefore, having now nothing to oppose its motion but the 
friction of the engine, is drawn to the top of the cylinder by the 
connt^^eight which it had raised during its down-stroke, and which 
is sufficiently heavy to raise the columns of water in the various 
pmnp mains, and to overcome the friction of the engine. As, how- 
ever, the piston would continue its motion until it came into contact 
with the cylinder cover, were it not checked, to do which the equi- 
librium valve G is closed before the piston has completed its up- 
stroke, after which the further motion of the piston, by compressing 
the steam on its upper side and attenuating that below it, occasions a 



386 THB ST£AM BKOINB. [PABT II. 

Bofficient preponderance of pressure on its upper surface to bring it 
gradually to a state of rest. The piston is now ready to resume its 
downward stroke as soon as the valve C shall be opened, and the 
steam from the boiler admitted to the top of the cylinder; this is 
effected at the proper time by means of a contrivance termed the 
cataract, by which the time which elapses between the up and down 
strokes can be regulated at the pleasure of the person who baa charge 
of the engine. The various contrivances by which the valves are 
opened are not shown in the plate, as it would require too long a 
description to render their use clearly intelligible ; it is sufficient to 
remark, that the valve C can be closed at any portion of the stroke 
which is desired, by means of a simple adjustment ; that the exhaus- 
tion valve F is so arranged as to close somewhat before the equi- 
librium valve G is opened ; that the injection cock of the condenser 
is only open during the same time that the exhaustion valve F is 
open ; and that the valve F is opened somewhat before the valve C, 
in order that the steam beneath the piston may be perfectly con- 
densed as soon as the down -stroke of the engine commences. 

We will now proceed to deduce the formula for calculating the 
effects of these engines, but since the nature of the work performed 
by the Cornish engine during its in-door and out-door strokes are 
so essentially different, it will be necessary to investigate each 
separately. 

First, then, during the in-door or down stroke, we have the pres- 
sure of the steam direct from the boiler upon the upper surface of 
the piston during the first portion of its stroke, which pressure will 
be so little less than that in the boiler itself, that without any appreci- 
able error it may be assumed to be the same, for the piston, engine 
beam, pump rods, and counterweight, comprising a joint mass weigh- 
ing several tons, require considerable force to put them in motion, 
and only acquire that motion very gradually, so that the steam has 
time to attam in the cylinder the same density and pressure as it 
had in the boiler; and during the after part of the stroke, we have 
the force due to the expansion of the steam above the piston. Now, 
if we put a for the area of the piston, x for the length of the cylmder 
equivalent to the clearance ana steam passages, p. for the pressure- 
in the boiler, l^ for the portion of the stroke pertormed before ex- 
pansion, and l^ for the whole length of the stroke from its com- 
mencement, we have from formula (XL*) 

for the whole amount of work performed during the in-door stroke 
of the engine; or, indeed, during both the in-door and out-door 
strokes, for the latter is, as already explained, performed by the 
descent of a counterweight which was raised during the in-door 
stroke ; and since the work expended in raising it is precisely equiva- 
lent to that which it performs during the out-door stroke, the 
amount which represents it would have to i^pear first on one ode of 



CHAP. VII.] THE 8T£AM BNOINB. 387 

the equation as a resistance, and afterwards upon the other as work 
performed, we shall therefore simplify the calculation hy omitting it 
altogether. 

Now, the actual resistances to which the Cornish engine is ex- 
posed are as follows; viz., during the in-door stroke* the pressure on 
the under side of the piston arising from the imperfect vacuum, 
which we will as hefore designate hy v; the friction proper to the 
motion of the engine when unloaded, equal /*; and the additional 
friction caused hy the load of the engine, represented by ^ ^j during 
the down stroke, and ^^^ during the up stroke, ^ having a different 
value in consequence of the different load upon the engine, and ^ 
being the effective load measured per square foot of the piston's 
surface: there is also a slight resistance occasioned by the steam 
pressing with rather more force upon the upper surface of the piston 
than the lower, in consequence of its having to pass through the passage 
and equilibrium valve, but this difference of pressure is too small to re- 
quire being taken notice of; the compression of the steam, however, 
in the upper part of the cylinder at the conclusion of the out-door 
stroke, and after the equilibrium valve has been closed, is of more 
consequence, and requires to be considered. If, then, we put p^ 
for the pressure of the steam on each side of the piston at the 
moment that the equilibrium valve is closed, p^ for the pressure of 
the steam above the piston at the conclusion of the out-door stroke, 
and /, the length of the stroke performed after the valve is closed^ 
we have from equation (IX.), 

or, sabstitnting for />,, in the last equation, its value from the first. 



{,^(^-)}-7 



Then, if / represent the length of the piston's out-door stroke 
remaining to be performed at any moment, and p the pressure of 
the steam above the piston at the same moment, we have from 
equation (IX.), 

tnd for the work performed in the elementary space A/, we have 

apAly 

in which, substituting the above value of /?, and taking the integral 
between the limits of / ss 0> and / =: Z,, we have 

z 



338 THE 8TBA1I BNOINE. [PABT II. 

p^ dl in \ n /-•» 

= ax(^ +>'0 ^^^'' ^^^^1( 7*'-' 

which hccomee, hy Bubstitating for p^ its value above in terms of/?,, 

which expression equals the work required to be done by the piston 
in compressing the steam in the upper part of the cylinder, after the 
equilibrium valve is closed. But a portion of this work is perfonned 
by the expansion of the steam on the under side of the piston, and 
therefore we must deduct this from the above. Now, the pressure 
of the steam on the under side of the piston at any moment being 
represented by />, and the length of the piston's stroke from its 
commencement at the same moment b^ng represented by I, we have 

and for the work performed in the elementary portion of the stroke 
represented by A /, we have ap A /, in which, substituting the abov& 
value of py and integrating between the limits of I z= l^^ and^ 
/ = /j "— ^3> we obtain 

r substituting for p^ its value above, 

a(/.-^,H-X),fJ^,(-; 4-;>.)hyper. log ^±^±-+1.!^ 

equal the work which the steam below the piston performs by ^l_ ts 
expansion after the equilibrium valve is closed; and therefore t^^fc® 
resistance actually to be overcome by the engine equals 



i 



CHJU>. VII.] THB 8TXAM BNOINB. 889 

Therefore, the total resistance during both strokes of the piston, 
equals 

2Pa/, = a4 {e(l + (p, 4. (p,) + T + 2/) 

- (/, - /« + A) hyper, log ^-i^-i-^ J ; 

and therefore, from the necessary equality between the work and 
the renstance, we have 

= a/,{e(l +*, + «),) + »+«/} 

-(/,-/3 + X) hyper. Iog^A+i_J. 

Now, since the Cornish engine only performs work in pumping 
during its in-door stroke, it is usual to take its effective Telocity 
as y/g = V, v being the number of double strokes which the engine 
mi^es per minute. Then, the quantity of steam actually consumed 
by the engine each double stroke, is only that which remains below 
the piston when the equilibrium valve is closed, and which equals 
a (/, — /, + X) ; and further, since the quantity of steam generated 
eqoak that consumed, we have 

1 V 

m wbich v, = , and y = - ; making which substitutions 

and solving in respect ofp^^ we obtain 

wUeh latter expression is the value of j^,, alreadv given at page 887. 
Then reducing, and solving in respect otp^^ we have 

_ /,Q /, -i-2X n 

^»-a^V(/,-/3 + X)- /, + x 7- 

z 2 



340 



THB 8TBAM BNOINB. 



[PABT II. 



to 

a 
'3 

a 
'^ 

2 



-I 

a 
► 
'So 

I 



08 O 

If- 

•I? 



•SIS 






p 

CO 

'■s 

2 
•a 



8l« 

+ 

+ 

I: 
+ 



+ 



> 






^ 



+ 



II 

Of 






? 



I: I: 



+ 




I I 



+ 



^ 


/< 


/< 


04 




04 


+ 


+ 


+ 


04 


•^ 




*J^ 




^* 



Is 

+ 






9 



/< 




/< 




X 


+ 


/< 


+ 


/< 


•f 


1 




1 




1 


-**• 


t 




t 


-^« 
o 


2 


o 


^ 








p 




i 




,£ 


3 


j: 


3 






II 












g^ 






+ 

I 



s 

04 

+ 

Is 

> 

I 



+ 

or 

H 
> 






CRAP. VII.] THE 8TBAM BNOINE. 341 

Single-acting Engine^ in which the steam presses only upon the 
lower surface of the piston. 

The engines belonging to this class are better known under the 
title of atmospheric engines, from the down stroke being performed 
entirely by the pressure of the atmosphere upon the upper surface 
of the piston. 

Plate XII, figure 4, is a section of the working cylinder of an 
atmospheric engine ; the steam at a low pressure enters the bottom 
of the cylinder A, by the valve B, and assists the counterweight at 
the other end of the beam in raising the piston. When a certain 
portion of the stroke has been completed, the valve B is closed, and 
the counterweight being insufficient of itself to overcome all the 
resistances of the engine, its velocity becomes gradually diminished, 
and the piston at last comes to rest at the top of the cylinder. The 
valve C is then opened, by which a communication is effected with 
the condenser; and a vacuum being formed under the piston, the 
pressure of the atmosphere upon its upper surface causes it to 
descend. In order to prevent the piston making too long a down 
stroke and coming in contact with the bottom of the cylinder, the 
valve C is closed some time before the conclusion of the stroke, and 
the oncondensed steam then remaining in the cylinder being com- 
pressed below the piston, gradually brings it to a state of rest. 

The work performed by the steam during the up stroke will be 
represented by formula (XL*) and if we put n for the pressure of 
the atmosphere, that performed during the down stroke will be 
lla/g, therefore we have 

« (^ + Pi) {^1 + (^1 + X) hyper, log ^±^^ - %/, + na/^ 

for the whole work performed during both strokes of the engine. 

Then we have for the resistance resulting from the friction of the 
engine when moving without any load, 2/2/9 and for the additional 
friction caused by the load, together with the resistance of the load 
itself, /^eO +?i + ^s)' ^^^9 finally, for the resistance caused by 
the imperfect vacuum, if we put l^ for the length traversed by the 
]nston after the valve C is closed, we have for the resistance before 
the closing of the valve, a n- (/j — l^) ; then if we put p for the 
pressure of the uncondensed steam, at any moment when the piston 
18 at a distance equal to / from the end of its stroke, we have from 
equation (IX.), 

l-\-\\q ' "J q 
whenee the work to be performed will be represented by 



^_ /3-f X /n \ n 



•<'.-<fTr(i-)-"l/^' 

= a(/,+X)(^ + ir) byper. log ^±^_2a/,. 



« (^3 + ^) f - + » j hyper, log 



342 THB BTBAM ENGINE. [p 

Therefore, collecting these several resistances, we have 

T w I uypcr. lUK — 

-la/, +a^,{f(l+^, +f,) 
+ a* (/,-/,) = a ( ^ + jP. ) |/, + (/, +'X) hyper, log ^ 

Now, in order to find the quantity of steam consnmed 
engine, we most deduct the quantity left below the piston w! 
yalVe C is closed, from the quantity admitted through the ' 
during the early portion of the stroke; the latter equals a{ 
with Uie pressure p^, and the former a (/^ -|- X) with the prea 
then, in order to find the space (equal S) which it would oc 
tha pressure />|, we have from equation (IX.) 

and solTing in respect of 8^ we have 

B = a(/, + X)!L±i^. 

And therefore the quantity consumed per minute equals 

,a{(/.+X)-(/,-.X)l±i^} = Q,.; 

then, substituting for v and v^ their yalues as before, redact 
solving in respect of />j, we have 

'^'~«yv(/. + x)^/j-|.x\,, ^ '; ?' 

Then, substituting this value of p^ in the preceding ei 
reducing, and solving in respect of V, Q, and f a V, we obtain 



GHAP. Vll.] THE STEAM ENGINE. 343 



V- 



<<.v- 



•"*'.+«{'-'-¥(=+')[,Ti-.+''— -mi! 

' Tbe remark made above with regard to the Cornish engine, of the 
want of experiments from which to deduce the values of the several 
resistances, will equally apply to tbe atmospheric engine, and we are 
therefore obliged to leave the above formula in their general form. 

PRACTICAL APPLICATION OF THE FOREOOINO FORMULA. 

In the preceding investigation we have deduced for each engine 
three forms of expression, giving the velocity of the piston in feet 
per minnte or V, the quantity of water in cubic feet evaporated by 
the boiler per minute or Q, and the effective useful work performed 
by the engine per minute expressed in pounds raised through a 
height of one foot or ^aV ; it may, however, be desirable, before 
concluding the subject, to deduce ^om these, expressions for the 
horse power, and t¥e quantity of coals consumed. 

In estimating the horse power of a steam engine, it is usual to 
assume the power of a horse to be equivalent to raising 33,000 lbs. 
through a height of one foot in. a minute; and, therefore, to derive 
the horse power of an engine whose effective useful work per minute 
equals ^aV, we have 

^gy _^ J the number of horses' power 
33000 ""I of the engine. 

Then, if we put F for the weight in pounds of coal, which is 
required to evaporate a cubic foot of water, we have 

^ r the number of pounds of coal consumed 

^ I by the engine per minute, 

QF J the number of pounds of coal consumed 
and 33000 — - = | p^^ horse power. 

The following table contains the value of F for four different 
varieties of coal, compared together by Mr, Wicksleed. 



344 



THB STEAM BKOINB. 



[part II- 



Description of Coal. 


Na of Ibf . of Coal 

required to evaporate 

one cubic foot of 

water «» F. 


Welsh coal, used in Corawall . . 
Best ditto, used at Old Ford . . . 
Anthracite, ditto . . . 
Best Newcastle small, ditto . , . 
Derbysliire coals, ditto . . . 


6-7414 
6-5838 
6-9337 
7-3322 
9-2292 



TABLE I.— Quantity of Coals equivalent to the horse power or 
33,000 lbs. raised one foot per minute in high pressure steam 
engines, when the greatest possible effect is obtained *. 



2S4 5 
251 
275 
21*2-8 
307 7 
3202 
343*1 






Is 



45 
60 

m 

180 
210 



^11 
II- 



74 

U'B 

29-7 
44-5 

142 

104 



Quant itr *nf maJ equfrAksi ta 
ooe bone potter* 



Foun<ls rai&fil otic fopi hi|^h «^i^- 
r Aleut to th^ Immediate pover of 



When working 
■I full pivHiire,. 



480 
1*>3 
08 
82 
74 
70 
6S 






143 
77 
&0 
51 
48 
414 



mt fulL pi^MUFe. 



llB. 

2,70a,ooo 

8,200,000 
13,700,000 

ie;,@oo,ow 

18,000,000 
10,200,000 
20,BOO,OQO 



Wbeti «oTlU.fi|| 



9,30a,000 

17,700,000 

33,7041,000 
26,2i>O,OO0 

2g,704i,a(ia 

32,200,000 



k 



TABLE IL — Quantity of Coals equivalent to the horse power om 
33,000 lbs, raised one foot per minute in condensing steam engines^ 
when the greatest possible effect is obtained. 



t 


c^ 


ll^ 




PcKiQdf raiwd one fofil hift* egml- 


^ 




s. g 


^M s Qtiant ity of €Oft] eaaivaJcnt to 


¥sk9tt to thfr tUttwft'Btr paver (x^ 


^mcr^ 


fl 


ll 


£ *B. <iOm hone 


power. 


thw cteim produ 

Efil 


»ct by 8« lli^ OM 


^ Mif 






tL 






Wh*n acbog 






1 S worn warning 
^ §'3 9.% full fireMUre* 


WhffD iForiLlnia 




H 




Eipuultely^. 


Bt Ml proiun. 


npuulrtJjt 




i- 


II1C}1I?I. 


ItM. 


Itii. 


Ibt. 


Ibi. 


Ibi. 




2^0 


35 


2-5 


63| 


40i 


2l,<KM3,0OO 


33,100^000 


"* J 


2^4-5 


45 


7-4 


63 


38* 


21,400,000 


35,200,000 




2fil'0 


00 1 


14-8 


60 


35! 


22,400,000 


37.500,000 


^ / 


275i) 


»0 1 


207 




331 




40.000,000 


/ 


292 8 


120 


445 




32i 




41,000,000 


/ 


307 7 


150^ 1 


593 




32 




42,400,000 


/ 


a^2 


ISO 


74 2 




314 




42,700,000 


/ 


3436 


240 


104 




at 




43,500,000 


J 



* The tables here given, marked I. II. III., were extracted, with the av 
thor*! peroiisrion, ^m Mr. Tredgold*8 work on Railroads. 



CHAP. Vll-I 



THE 8TB1M ENOINB. 



345 



Remarhon Tobies I. and II. — The colamns showing the pounds 
in engme oug^t to raise one foot high, by the heat of one bnshel of 
coals, are added chiefly for the pnrpose of comparison with actual 
practice. Now, it is stated, that after the most impartial examina- 
tion for seTeral years in succession, it is found that Woolf's engine, 
at Wheal Abraham Mine, raised 44,000,000 pounds of water, one 
foot high, witb a bushel of coals. And, ^^ the burning of one bushel 
of good Newcastle or Swansea coals in Mr. Watt's reciprocating 
engines, working more or less expansively, was found, by the 
aeconnts kept at tbe Cornish mines, to raise from 24,000,000 to 
32,000,000 pounds of water one foot high; the greater or less 
effect depending upon the state of the engine, its size, and rate of 
working, and tbe quality of the coal." 

We shall further add the results of half a year's reports taken, 
without selection, from Lean's Monthly Reports on the work per- 
fonned by the steam engines in Cornwall, with each bushel of coals. 
The numbers show the pounds of water raised one foot high with 
etch bushel, from January to June, 1818. 



Name of 



Pounds raised one foot by a bushel of coals. 



January. 



February. 



March. 



April. 



May. 



June. 



Stotf ComnMo Bb- 

|WW VOT <Woolf% 

^WtedAbnlMm (ditto) 
.Ditto (ditto) 

■Whad Unity (ditto) 
'Daloomh Cneine 
WlMl Abraham Engine 



iUiilsd Mines Cngj 
iTicskiiby Encine 
[WhsalCfaScgEn 



Engine 



>J8B,000 

ao.834,000 

41JM7.000 

314X10.000 
42^89,000 

aa,sw,ooo 

38.306,000 
38,733.000 
98,406,000 



SS.4S4,000 

26,108.000 

3S.364,000 
98,000.000 
393)6.000 
41.364.000 
36,180,000 
31.830,000 
30,375.000 
38,319.000 



91,808,000 29,969.000 
99,611,000 26,064.000 



23,606.000 



23,836.000 



30.445,000 
96.978,000 

40.490.000 
35,715.000 
31.427.000 
41.867.000 
33.594.000 



39.723.000 
23.e86.r" 



,000 29, 



29,032.000 30.336.000 

31 .520,000 '34.352,000 
1,702.000 '34.846.000 



41.888.000 
33,934,(NN) 
33.564,000 
41.823.000 
33,932,000 



38.233.000 '38.143.000 
33.714.000 ,34.291 .0(N» 
33.967.000 '30.105.000 
40.615.000 '42,098.<NIO 
— 35,797.000 



Tliese numbers are less than the immediate power of the engines, 
bj the friction and loss of effect in working the pumps ; hence, in 
comparinff them with Mr. Tredgold's table, it may be inferred that 
be msde bis calculations from such data as can be realised in practice. 
It is known from experience, that a cubic foot of water can be con- 
verted into steam equal in force to the atmosphere, with 7 lbs. of 
Newcartle coals; but we also know the attention necessary to pro- 
dace that effect, and therefore have assumed that %-f^ lbs. will be 
wjoired for that purpose. 

Aeeording to Mr. Leans Monthly Report, for January, 1 833, the 
foDowing engines raised more than 50,000,000 pounds, one foot 
* r consuming % bushel of coals : — 



846 



THE 8TBAM SNOINB. 



[PABT II. 



Name of Mine. 



Diameter of 
cylinder. 



Pounds raised 

bya 
bushdofoMls. 



Strokes 



Cardrew Downs . . . 
Binner Downs . . . . 

Ditto 

Consolidated Mines . . 

Ditto 

Ditto 

Ditto 

Ditto 

Ditto 

Polgooth 

Pembroke 

East Crinnis 

Wheal Leisure . . . . 

Wheal Vor 

Ditto 

Ditto 

Poladras Downs . . . 

Great Work 

Wheal Towan . . . . 

Ditto 

Wheal Falmouth Consols 
Wheal Darlington . . . 



66 inches 

70 

65 

90 

70 

65 

90 

90 

65 

66 

40 

76 

66 

80 

80 

53 

70 

60 

80 

80 

70 

80 



single 



51,831,751 
57,942,435 
55,931,852 
51,713,913 
61,846,133 
54,726,957 
59,978,983 
52,040,672 
65,617,011 
70,240,452 
61,170,237 
62,097,533 
53,506,372 
88,504,900 
65,471,147 
53,938,177 
56,766,668 
65,460,248 
73,159,628 
68,782,390 
54,334,137 
66,058,518 



734 
1013 
8-50 
4-86 
5-93 
8*60 
4*69 
6-50 
5-28 
800 
6-67 
704 
358 
6-22 
607 
6-37 
8-78 
608 
4-90 
6-98 
6-22 
8-47 



Of the above, the engine of greatest operation, the first at Whesl 
Vor, raises the water 190 fathoms, ^i seven lifts, drawing perpen- 
dicularly 160 fathoms, and the remainder diagonally. Main beam 
over the cylinder ; stroke in the cylinder ten feet ; one balance-bob 
at the surface, and three under ground. 

The following extract from Leans Monthly Report, for June 1841, 
will show the advance which had been then made. 



Name of Mine and Engine. 



Wheal Darlington, Eastern . . 
Great Wheal Fortune, Wheal 1 
Prosper ........ J 

Duffield 

WhealJulia. . . . 
Godolphin, Sims* . . 
Great Work, Leeds* . 
Wheal Vor, Borlase's . 
Ditto, Trelawny*s 
North Roskoar, New . 
Wheal Unity Wood, WUliams's 
Hallen-Beagle 



Diameter of 
cylinder, 
in indies. 



80, single 



80 
80 
80 
60 
80 
80 
70 
80 
70 



Poundi lifted 
one foot bra 
bushel of ooala. 


No.ofsti6kci 
perminut& 


81,681,776 


9-76 


64,169,466 


6-46 


64,438,341 


6-68 


60,966,983 


6-77 


60,166,186 


630 


66,776,208 


6-70 


74,927,176 


6-90 


60,634,127 


6KK> 


67,364,238 


4-76 


61,168,649 


6-64 


62,314,765 


6-60 



CHAP. VII.] 



THB STBAJi BNOINB. 



347 



Nun* oi Mine «iid Ea^am. 


D1uD«leror 


Pftundi lifted 
Ddv foot by » 
buBbti<frcc*Ift. 


^fo.flflt^a1tM 
pcrmLtiiitew 


Wheal UnitT 




70, tingle 

7a M 

50, combined 
85, kitig]« 

m> „ 

30 „ 
B5 „ 
85 „ 

50- .. 

ao „ 

80 „ 
66 „ 


08,462,34 fi 

70,(J35,787 

88,096,178 
9«,659,570 

55,219,358 
61,652,653 
€7.044, 127 

a,i3i,7ei 

54,477,451 
59,2tf7,244 
77,8M,927 
71,144,002 


6-70 
3-60 
4-30 
4-00 
B-40 
1030 
7i)7 
660 
3-65 
436 
530 
5W 


Catq Br«ft . , , . . 
INtto, Siou' . * * 
United Miiiei, Tajlori' * 
DitUi, OurdoEo'i . 
Ditto, Elfbn's , 
Ditto, Loam'i . , 
Ditto, Hocking'i 






United HilJif WiUiami^f . 
Fowef CcHiaolfj Aa»tin*f . 
PfllirMitll 










TABLE in.— /S^oiriny r^ gfecto of a force of traction of 100 Ibt. 
at different velocities^ on Canals^ Railroads^ and Turnpike-roads *. 
{From Tredgold,) 



TdodlTOflfcieloA. 


Lo«d nwjvtd by « power of 100 llw- 




•HBIlll. 


On » C»a»l- 


On » l»Td lUiliny, 


Onftln«l 
TuTDpDw Roadi 


mus 

DTMved, 




Total 
ErH>vddt 


tllfert. 


Tot*I 

IDAtl 




r 

5 
6 
1 
8 

10 
13-5 


366 

4-40 

5-13 

1 6-86 

7-33 

8^80 ' 

10-26 

11 -73 

13-20 

14-66 

10-0 


Ibi. 

55,500 

38,542 

Si8,310 

21,060 

,13,875 

9,635 

7,080 

5,420 

3,468 
1,900 


Ibt. 

39,400 

27,361 

20,100 

15,890 

9,850 

6,840 

5,026 

3,848 

3,040 

2,462 

1,350 


Itat 

li,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 
14,400 


lb*. 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 
10,800 


1,800 

1,800 
1,600 
1,800 
1,800 
1,800 
1,800 
1,800 
1,800 
1,800 
U800 


1,350 
1,350 

1,350 
1,350 
1,350 
1,3150 
1,360 
1,350 
1,360 
1,350 
1,350 



This table is intended to exhibit the work that may be performed 
by the same mechanical power, at different velocities, on canals, rail- 
roads,' and tampike-roads. Ascending and descending bj locks or 
canals^ may be considered equivalent to the ascent and descent of in- 
clinations on railroads and tampike-roads. The load carried, added to 
the weight of the vessel or carriage which contains it, forms the total 

• Though the force of traction on a canal varies as the square of the velocity, 
the me c h anical power necessary to move the boat is usually reckoned to increase 
as the cube of the velocity. On a railroad, or turnpike, the force of traction 
is constant; but the mechanical power necessary to move the carriage inereases 
as the vdodty. 



348 8TRBN0TH OF ANIMALS. [PART II. 

mass moved ; and the useful effect is the load. To find the effect on 
•canals at different velocities, the effect of the given power at one 
velocity being known, it will be as 3' : 2*5^ : : 55,500 : 38,542. 
The mass moved being very nearly inversely as the square of the 
velocity ; at least, within certain limits. 

This table shows, that when the velocity is five miles per hour, it 
requires less power to obtain the same eflfect on a railway than on a 
eanal ; and the lower range of figures is added to show the velocity 
at which the effect on a canal is only equal to that on a turnpike- 
road. By comparing the power and tonnage of steam vessels, it will 
be found that the rate of decrease of power by increase of velocity, 
is not very distant from the truth ; but we know that in a narrow 
canal the resistance increases in a more rapid ratio than as the square 
of the velocity *, that is, within certain limits ; beyond them, there is 
a remarkable change in the circumstances of resistance. 



Sect. IV. Animal Strength as a Mechanical Agent. 

1. The force obtained through the medium of animal agency, evi- 
dently varies, not only in different species of animals, but also in dif- 
ferent individuals. And this variation depends, first, on the particular 
constitution of the individual, and upon the complication of causes 
which may influence it ; secondly, upon the particular dexterity ac- 
quired by habit. It is plain, that such a variation cannot be sub- 
jected to any law, and that there is no expedient to which we can 
have recourse but that of seeking mean results. 

Secondly, the force varies according to the nature of the labour. 
Different muscles are brought into action in different gestures and 
positions of an animal which labours ; the weight itself of the ani- 
mal machine is an aid in some kinds of labour, and a disadvantage in 
others ; whence it is not surprising that the force exerted is different, 
in different kinds of work. Thus the force exerted by a man is 

* Aooording to the interestinff researches of Du Baat, the retistaDoe to the 
motion of boats, even in canals, may be regarded as proportional to the 
square of the velocity, or R as V* neariy, provided R be made to depend upon 
the transverse sections of the vessel and the canal in which it movea. If c be 
the vertical section of the canal, and b the vertical section of the immersed por. 
tion of the boat, or barge ; then 

The mean of Dn Buat'i ezperimmiu givM, K = 8'46, or 

^ - 8-46 ' 
but these experiments were not so numerous and varied as might be wished. 
See PrincipeM d* HpdrauHque^ tom ii. pp. 340, 342, &c. 



CHAP. VII.] 8TRBNGTH OF ANIMALS. 349 

different, in carrying a weight, in drawing or pushing it horizontally, 
and in drawing or pushing it vertically. 

Thirdly, the force varies according to the duration of the lahour. 
The force, for example, which a man can exert in an effort of a few 
instants, is different from that which he can maintain equahly in a 
course of action continued, or interrupted only hy short intervals, for 
a whole day of lahour, without inducing excessive fatigue. The 
former of these may he called Absolute Force j the latter Permanent 
Force, It is of use to hecome acquainted with them hoth, as it is 
often advantageous to avail ourselves sometimes of the one, some- 
times of the other. 

Lastly, the force varies according to the different degrees of velo- 
city widi which the animal, in the act of labouring, moves either its 
whole body, or that part of it which operates. The force of the 
animal is the greatest when it stands still ; and becomes weaker as 
it moves forward, in proportion to its speed ; the animal acquiring, 
at last, such a degree of velocity as renders it incapable of exerting 
any force. 

2. Let ^ be a weight equivalent to the force which a man can 
exert, standmg still : and let V be the velocity with which, if he pro- 
ceeds, he is no longer capable of exerting any force : also, let F be a 
weight equivalent to the force which he exerts, when he proceeds, 
equably, with a velocity v. 

Then F will be a functiou of v, such that, 1st, it decreases whilst 
V increases ; 2nd]y, when » = 0, then F = ^ ; 3rdly, when v = V, 
F = 0. 

3. Upon the nature of this function, we have the three following 
suppositions. 

1. Fs^fl— -V (Bouguer, Man. dee Vais.) 

2. Feaf^i—ljY (Euler, Nov. Comm. Pet. tom. IIL) 

3. Frs^Tl— M. (lb. tom. VIII ; and Act of Bowers.) 

4. CoroH. 1 . The effect of the permanent force being measured 
by the product F v, the expression for the effect will be one of the 
three following, accordingly as one or other of the suppositions is 
adopted. 

1. Fv(l-|),or^«(l-|.). 

^- ^^ a/('- !)'"*"('- S)- 

3. Pv(l.-^!),or^. (l-^y. 



350 



ANIMAL 8TRBNQTH : SCHULZB's BZPBRIMBNT8. [PART II. 



5. Coroil. 2. To know the weigbt with which a man shoold be 
loaded, or the velocity with which he onght to moTe, in order to pro- 
dace the greatest effect, we most make ^ . Fo e=s 0. 

Whence we shall haye 



1. F = - ^; and v = 

2 

2. F = - ^ ; and v = 



3. 



4 



>/3 

and I? = - V. 
3 



V = 0-5773 V. 



6. CoroB, 3. And the yalae of the greatest effect, according to the 
seyeral hypotheses, will be. 



1. \^y: 



2. 



3 >/3 



f V s 0-3836 f V : 



3. 27*^- 

Bat which of the three suppositions ought we to prefer ? And sre 
we certain that any of them approximates to the true law of nature? 

Mr. Schuize made a series of experiments with a view to the de- 
termination of this point*, and with regard to men decided in favour 
of the last of Euler s formulae : viz. 



,=,(.- 1)-. 



As the experiments of this philosopher are very little known in 
England, I shall here present hss brief account of them. 

7. To make the experiments on human strength, he took at rao* 
dom 20 men of different sizes and constitutions, whom he measured 
and weighed. The resnlt is exhibited in the following table. 



Order. 


Height. 


Weight. 


Order. 


Height 


Weight 


1 


5/ 3// 4W 


122 


11 


5/ 9// 7/// 


132 


2 


6 2 3 


134 


12 


6 1 4 


157 


8 


6 7 2 


165 


13 


6 8 2 


175 


4 


6 6 


181 


14 


5 4 1 


117 


5 


6 11 2 


177 


16 


5 10 8 


192 


6 


6 4 


158 


16 


6 8 


133 


7 


6 8 3 


180 


17 


4 11 2 


147 


8 


6 2 1 


117 


18 


6 8 9 


124 


9 


6 4 8 


140 


19 


5 6 


163 


10 


6 4 


126 


20 


6 10 1 


181 



* Mem. Acad. Sdcnc. Bsrlin, for 1783. 



V^ll.] ANIMAL araKNOTH: SCHULZB's BXPBRIMENT8. 



851 



e the heights are expressed in feet (marked ^), inches C')y 
m ("')y the feet being those of Rhinland, each 12-35 English 
The weights are in pounds, which are to our ayoirdupois 
30 to 29. 
ind the strength that each of these men might exert to raise a 

yerticaily, Mr. Schnlze made the following experiments : — 
took Tarious weights increasing bj 10 lbs. from 150 lbs. up to 
B. ; all these weights were of lead, having circular and equal 

To use them with suodess in the proposed experiments, he 
the same time a kind of bench made, in the middle of which 
hole of the same size as the base of the weights : this hole was 
y a circular cover when pressed against the bench ; at other 
t was kept at about the distance of a foot and a half above the 
bj means of a spring and some iron bars. To prevent the 
. with which this cover was loaded during the experiment from 
; down the cover, lower than the level of the surface of the 
he had several grooves made in the four iron bars, which sus- 
the cover, and which at the same time served to hold up the 
at any height where it might arrive by the pressure of the 

I as soon as the pressure of the weight ceased. 

Mr having laid the 150 lbs. weight on the cover, and the other 
s in succession increasing by 10 lbs. up to 250 lbs., he made 
[lowing experiments with the men whose size and weight are 
ibove, by making them lift up the wights as vertically as pos- 

II at once, and by observing the height to which they were 
i lift them. The annexed table gives the heights observed for 
Fereat weights mariced at its head. 



BO 


IGO 


170 


180 


194 


200 


iio 


223U 230 


240 


MO 


m 


*f m 


it m 


ri m 


jtf m 


U Hi 


it tit 


** H* if aif'i 


ff Iff 






B 4 


4 11 


4 4 


a a 


2 B 


1 1 


I 






10 


6 6 


5 7 


4 7 


3 u 


2 5 


6 


17 3 








7 3 


6 5 


a 9 


4 11 


4 


3 , 


3 8 3 1 


1 4 






7 6' 


7 2 


d 10 


5 3 


4 7 


4 


4 7 3 2 


1 3 






M 1 


9 7 


8 5 


7 10 


7 1 


5 10 


6 6 4 1 


1 






U 


13 5 


12 8 


n 


10 I 


8 a 


3 8 I 11 


2 




11 


11 3 


10 5 


9 3 


8 1 


6 9 


& 3 


6 J 3 2 


1 






10 s 


4 


8 11 


a 1 


« 11 


h SO 












8 3 


7 1 


5 fl 


4 I 


3 


1 3 












« b 


4 7 


3 g 


2 5 


1 7 


4 











B table proves that the size of the men employed to raise the 
ts Terdcally has considerable influence on the height to which 
ronght the same weight. We find also that the height dimi- 
in a much more considerable ratio than the weight increases ; 
e may therefore conclude, that it is advantageous to employ 
nen when it becomes necessary to draw vertically from below 
ds : and on the contrary, it is more advantageous to employ 
>f a considerable weight, when it is required to lift up loads by 



352 



ANIMAL STBBNOTH: SCHULZE's EXPERIMENTS. [PABT II. 



means of a pulley about which a cord passes, that the workmen may 
draw in a vertical direction, from above downwards. To find the 
absolute strength of these men in a horizontal direction, Mr. Schulze 
proceeded thus : — 

Having fixed over an open pit a brass pulley, extremely well made, 
of 15 inches diameter, whose axis, made of well polished steel to 
diminish the friction, was ^ of an inch in diameter, he passed over 
this pulley a silk cord, worked with care, to give it both the neces- 
sary strength and flexibility. One of the ends of this cord carried a 
hook to hang a weight to it which hung vertically in the pit, whilst 
the other end was held by one of the 20 men, who in the first order 
of the following experiments made it pass above his shoulders; 
instead of which, in the second, he simply held it by his hands. 

Mr. Schulze had taken the precaution to construct this in such a 
manner that the pulley might be raised or lowered at pleasure, in 
order to keep the end of the cord held by the man always in a hori- 
zontal direction, according as the man wss tall or short, and exerted 
his strength in any given direction. 

He had made the necessary arrangements so as to be able to load 
successively the basin of a balance which was attached to the hook 
at the end of the cord which descended into the pit, whilst the man 
who held the other end of this cord employed all his strength with- 
out advancing or receding a single inch. 

The following tables give the weights placed in the basin when 
the workmen were obliged to give up, having no longer sufficient 
strength to sustain the pressure occasioned by the weight. To pro- 
ceed with certainty, Mr. Schulze increased the weight each time by 
five pounds, beginning from 60, and took the precaution to make 
this augmentation in equal intervals of time ; having always precisely 
a space of 10 seconds between them. The result of these observa- 
tions repeated several days in succession, is contained in the follow- 
ing tables. 

I. When the cord passed over the shoulders of the workmen : 



Order. 


lbs. 


Order. 


lbs. 


Order. 


Ibe. 


Order. 


lU. 


1 


95 


6 


100 


11 


95 


16 


95 


2 


105 


7 


115 


12 


100 


17 


100 


3 


110 


8 


105 


13 


110 


18 


90 


4 


100 


9 


95 


U 


90 


19 


100 


5 


105 


10 


90 


15 


110 


20 


100 



CHAP. VII.] ANIMAL STRENGTH: MEN. 

II. When the cord was simply held before the man : 



353 



Order. 


Ibt. 


Order. 


lbs. 


Order. 


lbs. 


Order. 


lbs. 


1 


90 


6 


100 


11 


90 


16 


90 


2 


105 


7 


no 


12 


90 


17 


90 


3 


105 


8 


100 


13 


100 


18 


85 


4 


90 


9 


90 


14 


85 


19 


100 


5 


95 


10 


85 


15 


105 


20 


100 



These two tables show that men have less power in drawing a 
cord before them than when they make it pass over their shoulders ; 
thej show, also, that the largest men have not always the greatest 
strength to hold, or to draw in a horizontal direction, by means of a 
cord. To obtain the absolute velocity of these twenty men, Mr. 
Schnlze proceeded as follows : — 

Having measured very exactly a distance of 12,000 Rhinland feet, 
in a plane nearly level, he caused these twenty men to march with a 
fair pace, but without running, and so as to continue during the 
period of four or five hours ; the following is the time employed in 
describing this space, with the velocity resisting for each of them. 



Order. 

1 


Time. 


Velocj 


Order. 


Time. 


Veloc 


Order. 


Time. 


Veloc 


4018 


4-94 1 


8 


4009 


4-99 ! 


15 


3617 


5-51 


2 


4112 


4-85 


9 


40-20 


4-96 1 


16 


41-28 


4-82 


3 


3908 


5-55 ' 


10 


40-51 


4-90 ' 


17 


42-25 


4-71 


4 


39-40 


504 


11 


3617 


5-51 


18 


40-19 


4-98 


5 


3419 


5-83 


12 


38-11 


5-24 


19 


39-57 


501 


e 


3511 


5-68 


13 


3805 


5-25 


20 


37-51 


5-29 


7 


3807 


5-25 


14 


3701 


5-40 









It is necessary to mention, with regard to these experiments^ that 
Mr. Schulze took care to place at certain distances persons in whom 
he could place confidence, in order to observe whether these men 
marched uniformly and sufiiciently quick without running. 

Having thus obtained not only the absolute force, but the absolute 
velocity also, of several men, he took the following method to deter- 
mine their relative force. 

He made use of a machine composed of two large cylinders of 
very hard marble, which turned round a vertical cylinder of wood, 
and moved by a horse, which described in his march a circle of 
10 Rhinlaod feet. This machine appeared the most proper to make 
the subsequent experiments, which serve to determine the relative 
strength that the men had employed to move this machine, and which 
is used hereafter to determine which of Euler's two formulas ought 
to be preferred. 

To obtain this relative force, he took here the same pulley which 



354 ANIMAL 8TRIN0TH : MBK. [PART II. 

served in the preceding experiments, by applying a cord to the 
vertical cylinder of wood, and attaching to the other end of this cord, 
which entered into an open pit, a sufficient weight to give suceseivelv 
to the machine different velocities. 

Having applied in this manner a weight of 215 lbs., the machine 
acquired a motion, which after being reduced to a uniform velocity, 
taking into account the acceleration of the weight, of the friction, 
and of the stiflfness of the cord, gave 2*41 feet velocity ; and having 
applied in the same manner a weight of 220 lbs., the resulting 
uniform motion gave a velocity of 2*47 feet. These two limits are 
mentioned because they serve as a comjmrison with what imme- 
diately follows : Mr. Schulze began these experiments with a weight 
of 100 lbs., and increased it by five every time from that number op 
to 400 lbs. 

He made this machine move by the first seven of his workmen, 
placing them in such a way that their direction remained almost 
always perpendicular to the arm on which was attached the cord 
which passed over their shoulders in an almost horizontal direction. 

Thus situated, they made 281 turns with this machine in two 
hourH, which gave for their relative velocity p = 2*45 feet per second. 
We have also the absolute force, or f, from these scTen men, 
by the above table, ^ 730 lbs. : and their absolute velocity or 
V = 5-30 feet. 

Therefore, by substituting these values in the first formola, we 
find the relative force F = 205 lbs., which agrees very well with 
what we have just found above. 

If instead of this first formula the second be taken, it gives 
F =: 153 lbs., which is far too little. 

By this it is evident, that the last of Ruler's two formulae is to be 
preferred in all respects. Mr. Schulze made a great number of com- 
binations, and almost alu'ays found the same effect. 

Dividing the 205 lbs. which we have just found by seven, the 
number of workmen, we get 29 lbs. for the relative force with 
2'45 feet relative velocity for each man, which is rather more than 
the values commonly adopted in the computation of machinery. A 
number of other observations on different machines have given the 
same result; that is to say, we must value the mean human strength 
at 29 or 30 lbs., with a velocity of 2|feet per second. 

To obtain the ratio of the strei^gth of a horse to that of a mm, 
Mr. Schulze proceeded in a similar manner; but his results, in 
reference to that inquiry, are neither so correct nor so interesting. 

8. In the first volume of my Mechania^ I stated the ayerage force 
of a man at rest to be 70 lbs., and his utmost walking velocity when 
unloaded, to be about six feet per second; and thence inferred that a 
man would produce the greatest momentum when drawing Sl^lbs* 
along a horizontal plane with a velocity of two feet per second. But 
that is not the most advantageous way of applying human strength. 

9. Dr. Desagaliert asserts, that a man can raise of water or any other 



OHAP. VII.] ANIMAL STRENGTH : MEN. 355 

weight aboat 550 lbs., or one hogshead (weight of the vessel included), 
ten feet high in a minute; this statement, though he says it will 
bold good for six hours, appears from his own facts to be too high, 
and 18 certainly such as could not be continued one day after another. 
Mr. Smeaton considers this work as the effect of haste or distress ; 
and reports, that six good English labourers will be required to raise 
21141 cubic feet of sea water to the height of four feet in four hours: 
in this case, the men will raise a very little more than six cubic feet 
of freah water each to the height of 10 feet in a minute. Now, the 
hogshead containing about 8^ cubic feet, Smeaton s allowance of work 
proves less than that of Desaguliers in the ratio of 6 to 8^ or 3 to 4|. 
And as his good English labourers who can work at this rate are 
ettiinated by him to be equal to a double set of common men picked 
Qp at random, it seems proper to state that, with the probabilities of 
voluntary interruption, and other incidents, a man's work for several 
neoessive days ought not to be valued at more than half a hogshead 
laised 10 feet high in a minute. Smeaton likewise states, that two 
ordinary horses will do the work in three hours and twenty minutes, 
which amounts to little more than two hogsheads and a half raised 
10 feet high in a minute. So that, if these statements be accurate, 
one horse will do the work of five men. 

Mr. Emerson affirms, that a man of ordinary strength, turning a 
roller by the handle, can act for a whole day against a resistance 
equal to 30 IbSs weight ; and if he works 10 hours a day, he will 
rsise a weight of 30 lbs. through 3^ feet in a second of time ; or, if 
the weight be greater, he will raise it to a proportionally less height. 
If two men work at a windlass, or roller, they can more easily draw 
up 70 lbs. than one man can 30 lbs., provided the elbow of one of the 
handles be at right angles to that of the other. Men used to bear 
loads, such as porters, will carry from 150 lbs. to 200 or 250 lbs., 
scGording to their strength. A man cannot well draw more than 
70 Iba. or 80 lbs. horieontally : and he cannot thrust with a greater 
force acting horisontally at the height of his shoulders, than 27 or 
80 lbs. But one of the most advantageous ways in which a man can 
exert bis force is to sit and pull towards him nearly horizontally, as 
in the action of rowing. 

M. CSoulorob communicated to the French National Institute the 
results of various experiments on the quantity of action which men 
csn afford by their daily work, according to the different manners in 
which they employ their strength. In the first place he examined 
the quantity of action which men can produce when, during a day, 
they mount a set of steps or stairs, either with or without a burden. 
He found that the quantity of action of a man who mounts without a 
burden, having only his own body to raise, is double that of a man 
loaded with a weight of 68 kilogrammes, or 150 lbs. avoirdupois, both 
eoDtinuing at work for a day. Hence it appears how much, with 
equal fatigue and time, the total or absolute effort may obtain different 
fdnes by varying the combinations of effort and velocity. 

But ihe word effect here denotes the total quantity of labour 

A A 2 



356 ANIMAL STRENGTH: MBK. [PABT IT. 

employed to raise, not only tbe burden, but the man himself; and, 
as Coulomb observes, what is of the greatest ]m|K)rtance to consider 
is the useful effect,, that is to say, the total effect, deducting the value 
which represents the transference of the weight of the man's body. 
This total effect is the greatest possible when the man ascends without 
a burden ; but the uteful effect is then nothing : it is also nothing 
if the man be so much loaded as to be scarcely capable of movbg, 
and consequently there exists between these two limits a value of tbe 
load such that the useful effect is a maximum. M. Coulomb supposes 
that the loss of quantity of action is proportional to tbe load (an 
hypotliesis which experience confirms), whence he obtains an equa- 
tion which, treated according to the rules of maxima and minima, 
gives 53 kilogrammes (117 lbs. avoirdupois) for the weight with 
which the man ought to be loaded, in order to produce during one 
day, by ascending stairs, the greatest useful effect : the quantity of 
action which results from this determination has for its value 5(^ kilo- 
grammes ( 1 23| lbs. avoirdupois) raised through one kilometre, or nearly 
1094 yards. But this method of working is attended with a loss of 
three-fourths of the total action of tlie man, and consequently costs four 
times as much as work, in which, after having mounted a set of steps 
without any burden, the man should suffer himself to fall by any means, 
so as to raise a weight nearly equal to that of his own body. 

From an examination of the work of men walking on a horizontal 
path, with or \%nthout a load, M. Coulomb concludes that the greatest 
quantity of action takes place when the men walk being loaded ; and 
is to that of men walking under a load of 58 kilogrammes (128 lbs. 
avoirdupois) nearly as 7 to 4. The weight which a man ought to 
carry in order to produce the greatest useful effect,, namely, that effect 
in which the quantity of action relative to the carrying his own 
weight is deducted from the total effect, is 50*4 kilogrammes, or 
111*18 lbs. avoirdupois- 
There is a particular case which always obtains with respect to 
burdens carried in towns, viz., that in which the men, after having 
carried their load, return unloaded for a new burden. The weight 
they should carry in this case, to produce the greatest efiect, is 
61*25 kilogrammes (135^ lbs. avoirdupois). The quantity of osefol 
action in this case, compared with that of a man who walks freely to^ 
without a load, is nearly as 1 to 5, or, in other words, he employs to 
pure loss ^ths of his power. By causing a man to mount a set of 
steps freely and without burden, his quantity of action is at leait 
double of what he affords in any other mediod of employing ^ 
strength. 

When men labour in cultivating the ground, the whole qoantitj 
afforded by one man during a day amounts to 100 kilogrammes ele- 
vated to 1 kilometre, that is, 220*6 lbs. raised 1094 yards. M. 
Coulomb, comparing this work with that of men employed to csrr; 
burdens up an ascent of steps, or at the pile-engine, finds i loM 
of about ^ th part only of the quantity of action, which may he 
neglected m researches of this kind. 



CHAP. VII.] ANIMAL STRBNGXH : IIBN. 357 

In estimating mean results we should not determine from experi- 
ments of short duration, nor should we make any deductions from the 
exertions of men of more than ordinary strength. The mean results 
baye likewise a relation to climate. '' I have caused," says M. Cou- 
lomb, ^ extensive works to be executed by the troops at Martinico, 
where the thermometer (of Reaumur) is seldom lower than 20^ 
or 77^ of Fahrenheit). I have executed works of the same kind by 
\e troops in France, and I can affirm, that under the fourteenth 
degree of latitude, where men are almost always covered with perspir- 
ation, they are not capable of performing half the work they could 
perform in our climate." 

10. Entirely according with these are the experiments of Regnier, 
by means of a dynamometer, the results of which not only established 
the superiority of civilized men over savages, but that of the English- 
man over the Frenchman. The following is reduced from one of 
Regnicr's tables of mean results. 



£ 



Strength. 



Savages, of Van Dicmen's Land 

New Holland . . . 

Timor 

Civilised men : French . . . 
English . . . 



With the 


With the 


hands. 


reins. 


Ibf. (m. 


lbs. OS. 


30 6 





51 8 


14 8 


58 7 


10 2 


69 2 


22 1 


71 4 


23 8 



11. A porter in London is accustomed tu carry a burden of 
200 Ibe. at the rate of three miles an hour; and a couple of chairmen 
continue at the rate of four miles an hour, under a load of 300 lbs. 
Yet these exertions, Professor Leslie remarks, are greatly inferior to 
the subanltory labour performed by porters in Turkey, the Levant, 
and generally on the shores of the Mediterranean. At Constan- 
tinople, an Albanian porter will carry 800 or 900 lbs. on his back, 
stooping forward, and assisting his steps by a short staff. Such loads, 
however, are carried for very short intervals. At Marseilles it is 
affirmed that four porters carry the immense load of nearly two tons, 
by means of soft hods passing over their hcads^ and resting on their 
shoulders, with the ends of poles from which the goods are sus- 
pended. 

12. With regard to the magnitude of the comparative efforts of 
man in different employments, the late Mr. Robertson Buchanan 
ascertained, that in working a pump, in turning a winch, in ringing 
a bell, and rowing a boat, die dynamic results are as the numbers 
100, 167, 227, and 248. 

According to the interesting experiments described in M. Hachette's 
Traiie des Mctckinesy the dynamic unity being Uie weight of a cubic 
metre of water raisid to the height of one metre [that is, 2208 lbs. 
avoirdupois, or 4 hogsheads raised to the height of 3*281 feet, or 



356 ANIMAL STRSlfeTH: lfEK» [PABT 

1*3124 hogsheads to the height of 10 feet], we have the folio? 
measures, at a medium, of the daily actions of men. 

Dyn. I 

1. A man marching 7| hours on a slope of 7 degrees 

with a load of from 15 to 18 lbs = 

2. Marching in a mountainous country without load = 

3. Carrier of wood up a ladder, his weight 123 Ibs^ his 

load llTlbs = 

4. Carrier of peat, up steps, his own weight compnced, 

112 to = 

5. Man working at the cord of a pulley to raise the ram \ 

of a pile engine: three examples ) 

6. A man drawing water from a well by means of a cord as 

7. Man working at a capstan := 

8. Man working at a capstan to raise water, mean of 24 

observations ! = 

The unit of transport being the weight of a cubic metre of w 
carried a metre (or 2208 lbs. carried 3281 feet) upon a horizo 
road, we have for the daily action, 

Dyn. T 

1. A man travelling without load on a flat road, his weight 

1 54 lbs. his journey 31 ^ miles ^ 3 

2. A soldier, carrying from 44 to 55 lbs., travelling 12| 

miles, 1800 to = 1 

3. Ditto, a forced march of 25 miles = 2 

4. A French porter, weight of the man not included, 

792 to =« 

5. Porter with wheel-barrow, weight of the man not 

included b: 1 

6. Porters with a sledge ss 

7. A man drawing a boat on a canal; 110310 lbs. coq- 

veyed 6 J miles * xr= 550 

14. Mr. B. Bevan, an able engineer, has made experiments on 
application of human energy to the use of augurs, gimlets, scr 
drivers, &c. He has presented to the public the following list, 
^)echnen ; premising that many ordinary operations are performe 
a short space of time, and may therefore be done by greater exei 
than if a longer time was necessary. Thus a person, for a a 
time, is able to use a tool or instrument called 

lbs. 

A drawing-knife, with a force of 100 

An augur, with two hands 100 

A screw-driver, one hand 84 

A common bench vice handle 73 

A chisel and awl, vertical pressure 72 

A windlass, handle revolving 60 



CHAP. VII.] ANIMAL STBBNOTH : MEN. 35^ 

lb«. 

Pincers and pHers, compression 60 

A hand-plane, horizontallj 50 

A hand or thamb-vice 45 

A hand-saw 36 

A stock-bit, revolving 16 

Small screw-drivers, or twisting by the 

thumb and fore-finger only 14 

15. M. Morisot informs us that the time employed by a French 
stone-mason's sawyer, to make a section of a square toise (40*89 
square feet English) in different stones, is as below : viz. 

hours. 

Calcareous stone, equal grain, spec. grav. 2000 45 

hard, spec. grav. 2300 62 

Liais, ditto hard, fine grain, spec. grav. 2400 67 

Pyrenean alabaster^ the softest of the marbles 56 

Normandy granite 504 

Granite from Vosges 700 

Red and green porphyry 1177 

The workmen ordinarily made 50 oscillations in a minute ; each 
stroke about 15| inches. 

16. Hassenfratz assigns 13 kilogrammes as the mean effort of such a 
man ; but M. Navier, in his new edition of BdidoTy Architecture Hydrau- 
Uque^ regards this estimate as too high. If Hassenfratz were correct, 
the daily quantity of action of the sawyer would be equivalent to 376 
kilogrammes elevated to a kilometre (or 818 lbs. raised g of a mile), 
a quail titv more than triple that of a man working at a winch. M. 
Navier gives, as a more correct measure of this labour for 1 2 hours, 
188 kilogrammes raised a kilometre : half the former measure. But 
«U this is probably very vague. 

17. Among quadrupeds, those which are employed to produce a 
mechanical effect are the dog, the ass, the mule, the ox, the camel, 
and the horse. Of these the horse is the only one, so far as we are 
aware, whose animal energy has been subjected to cautious experi- 
ments ; and, even with regard to this noble animal, opinions as to 
actoal results are very much afloat. The dynamic effort of the horse 
is, however, probably about 6 times that of. a strong and active 
labourer. Desaguliers states the proportion as 5 to 1, coinciding with 
the deductions of Smeaton. The French autliors usually regard 
sevep men as equivalent to one horse. As a fair mean between 
theae, I assumed in vol. i. of my Mechanics the proportion of 6 to 1, 
and stated the strength of a horse as equivalent to 420 lbs. at a dead 
pull. But the proportion must not be regarded as constant, but 
obviously varies much according to the breed and training of the 
animal, as well as according to the nature of the work about which 
he is employed. Thus the worst way* as De la Hire observed, of 



860 ANIMAL STRBNOTH: H0R8B8 [PABT II. 

applying the strength of a horse is to mal«c him carry a weight up a 
steep hill ; while the organization of a man fits him very well for tiiat 
kind of lahour: hence three men, climhing up such a hill with a 
weight of 100 Ihs. each, will proceed faster than a horse with a load 
of 300 Ihs. 

18. In the memoirs of the French Academy for 1703 are inserted 
the comparatiye ohservations of M. Amontons, on the velocity of 
men and of horses ; in which he states the velocity of a horse loaded 
with a man and walking to he rather more than 5| feet per second, 
or 3 1 miles per hour, and when going a moderate trot with the same 
weight to he ahout 8J feet per second, or ahout 6 miles per hour. 
These velocities, however, are somewhat less than what might have 
heen taken for the mean velocities. 

19. But the best way of applying the strength of horses is to 
make them draw weights in carriages, &c. To this kind of labour, 
therefore, the inquiries of experimentalists should be directed. A 
horse put into harness, and making an effort to draw, bends himself 
forward, inclines his legs, and brings his breast nearer to the earth ; 
and this so much the more as the effort is the more considerable. 
So that, when a horse is employed in drawing, his effort will depend, 
in some measure, both upon his own weight and that which he 
carries on his back. 

Indeed it is highly useful to load the back of a drawing horse to a 
certain extent; though this, on a slight consideration, might be thought 
to augment unnecessarily the fatigue of the animal : but it must be 
recollected that the mass with which the horse is charged vertically 
is added in part to the effort which he makes in the direction if 
traction, and thus dispenses with the necessity of his inclining so 
much forward as he must otherwise do : and may, therefore, under 
this point of view, relieve the draught more than to compensate for 
the additional fatigue occasioned by the vertical pressure. Carmen, 
and waggoners in general, are well aware of this, and are commonly 
very careful to dispose of the load in such a manner that the shaf)^ 
shall throw a due proportion of the weight on the back of the shall 
horse. This is most efficaciously accomplished at Yarmouth, in Nor- 
folk, where a number of narrow streets connecting the market-place 
i^dth the quay, have led to the invention and use of the low, strong, 
narrow carts, thence denominated Yarmouth carts, drawn by one 
horse; and on which the loads are frequently shifted, especially when 
the vehicles pass over the bridge, in order to give the animals better 
foot-hold, and consequently a greater dynamic effort. • 

20. The best disposition of the traces during the time a horse is 
drawing is perpendicular to the position of the collar upon his breast 
and shoulders : when the horse stands at ease, this position of the 
traces is rather inclined upwards from the direction of the road ; bat 
when he leans forward to draw the load, the traces should then be- 
come nearly parallel to the plane over which the carriage is to be 
drawn ; or, if he he employed in drawing a sledge, or any thing wiili 

• out wheels, the inclination of the traces to the road should (from the 



CHAP. Vn.'] ANIMAL 8TRKNGTH : HOftSIS. 361 

table at page 52QO) be about 18|% when the friction is one-third of 
die presfiure. If tlie relation of the friction to the pressure he dif- 
ferent from this, the same tahle will exhibit the angle which the traces 
most make with the road. 

21. When a horao is made to more in a circular path, as is often 
pnctised in mills and other machines moved by horses, it will bo 
necesMry to give the circles which the animal has to ^'alk round the 
greatest diameter that will comport with the local and other con- 
diuuns to which the motion roust be subjected. It is obvious, indeed, 
tLat, since a rectilinear motion is the most easy for the horse, the less 
the line in which he moves is curved, with the greater facility he will 
walk over it, and the less he need recline from a vertical |)osition : 
and besides this, with equal velocity the centrifugal force will be less 
b tbe greatest circle, which will proportionally diminish the friction 
of tbe cylindrical part of the trunnions, and the labour of moving the 
machbe. And, further, the greater the diameter of the horse- walk, 
tbe nearer the chord of the circle in which the horse draws is to coin- 
cidence with the tangent, which is the most advantageous position of 
tbe line of traction. On these accounts it is that, although a horse 
man draw in a circular walk of 18 feet diameter, yet in general it is 
advisable that the diameter of such a walk should not be less than 
25 or SO feet ; and in many instances 40 feet would be preferable to 
either. 

ii. It has been stated by Desaguliers and some others, that a 
borse employed daily in drawing nearly horizontally can move, during 
cigbt hours in the day, about 5iOO lbs. at the rate of 2i miles per 
hoar, or 3| feet per second. If the weight be augmented to about 
240 or 250 lbs., the horse cannot work more than six hours a day, 
and that with a less velocity. And, in both cases, if he carry some 
weight, he will draw better than if he carried none. M. Sauveur 
estimates the mean effort of a horse at 175 French, or 189 avoirdupois 
pounds, with a velocity of rather more than three feet per second. 
But all these are probably too high to be continued for eight hours, 
&T after day. In another place Desaguliers states the mean work of 
a horse as equivalent to the raising a hogshead full of water (or 550 
lbs.) 50 feet high in a minute. But Mr. Smeaton, to whose authority 
much is due, asserts, from a number of experiment*, that the greatest 
effect is the raising 550 lbs. forty feet high in a minute. And, from 
wne experiments made by the Society for the Encouragement of 
Arts, under the direction of Mr. Samuel Moore, it was concluded, 
that a horse moving at the rate of three miles an hour can exert a 
force of 80 lbs. Unluckily, we are not sufficiently acquainted with 
tlw nature of the experiments and observations from which these 
MnctioDs were made to institute an accurate com]iarison of their 
'«*Qlt». Neither of them ought to express what a horse can draw 
"pon a carriage ; because in that case friction only is to l)e overcome 
(«fter the load is once put into motion); so that a middling horse, well 
applied to a cart, will often draw nmcli more than 1000 lbs. The 
^^^ estimate would be that which measures the weight that a 



3G2 ANIMAL STBSKGTH: H0B8B8. [PABT n. 

horse wonld draw up out of a well ; the animal acting hj a horizontal 
line of traction turned into the yertical direction hj a simple pulley, 
or roller, whose friction should he reduced as modi as poesihle. 

23. Mr. Tredgold, in his valuahle puhlication on Railroads, has 
directed his attention to the suhject of *'*'hor9e power" The follow- 
ing is his expression for the power of a horse, 250 ^ \l ) ; and 

\ V / for the day's work in Ihs. nused one mile ; d 

1 -f « 

heing the hours which the horse works in a day, and the weight of 

14*7 

the carriage to that of the load as n : 1. He also gives -r for the 

a/ d 

greatest speed in miles per hour, when the horse is unloaded. These 
expressions must, at present, he regarded as tentative. The follow- 
ing is his tahlc of the comparison of the duration of a horse's daily 
labour and maximum of velocity, imloaded. 

Duration of labour. Max. Telocity unloaded 

Hours, in mile* per hour. 

1 147 

2 10-4 

3 8-5 

4 7-3 

5 6-6 

6 6-0 

7 6-5 

8 5-2 

9 4-9 

10 4-6 

Taking the hours of labour at 6 per diem, the utmost that Mr- 
Tredgold would recommend, the maximum of useful effect he assigns 
at 125 lbs. moving at the rate of three miles per hour, and regardiiif 
the expense of carriage, in that cose, as unity ; then — 

Miles per hour. Proportional expense. Moving fbroe. 

2 li or M25 166 lbs. 

3 1 125 

3J 1^7 or 1-0285 104 

4 li or 1125 83 

4j l| or 1-333 62j 

5 l| or 1-8 41$ 

5i 2 36J 

That is, the expense of conveying goods at 3 miles per hour bang 
1 ; the expense of 4| miles per hour will be 1^ ; and so on, the eX' 
pense being doubled when the speed is 5^ miles per hour. 

24. Thus, according to Mr. Tredgdd^ wc have for the day of 



CfiAK TU.] ANIMAL 8TB1N0TH *. HOBSES. 363 

6 hours 2d60 lbs. raised one mile. And Mr. Bevan^ who has made 
many experiments on the force of traction to move canal boats on the 
Grand Junction Canal, found the force of traction 80 lbs., and the 
space travelled in a day 26 miles ; hence, it is only equivalent to 
26 X 80 = 2080 lbs. raised one mile for the day's work ; the rate 
of travelling being 2*45 miles per hour ; and the result a little less 
than Mr. Tredgdd'Sy the difference probably arising from the devia- 
tion of the angle of the catenary formed by the rope from the horizon. 
25. The following experimental data from Mr. Bevan also deserve 
attention. 

** In the period from 1803 to 1809, I had the opportunity of as- 
certaining correctly the mean force exerted by good horses in draw- 
ing a plough ; having had the superintendence of the experiments on 
that bead at the various ploughing matches both at Woburn and Ash- 
ridge, nnder the patronage of the Duke of Bedford and the Earl of 
Bridgewater. I find among my memoranda the result of eight 
ploughing matches, at which there were seldom fewer than seven 
teams as competitors for the various prizes. 

lbs. 
The first result is from the mean force of each horse in six 

teams, of two horses each team, upon light sandy soil = 156 
The second result is from seven teams of two horses each 

team, upon loamy ground, near Great Berkhampstcad... = 154 
The third result is from six teams of four horses each team, 

with old Hertford^ire ploughs =: 127 

The fourth result is from seven teams of four horses each 

team, upon strong stony land (improved ploughs) = 167 

The fifth result is from seven teams of four horses each 

team, upon strong stony land (old Hertfordshire ploughs) := 103 
The sixth result is from seven teams of two horses each 

team, upon light loam = 177 

The seventh result is from five teams of two horses each, 

upon light, sandy land = 170 

The eighth result is from seven teams of two horses each 

team, upon sandy land = 160 

" The mean force exerted by each horse from fifty-two teams, or 
one hundred and forty-four horses, equals 163 pounds each horse, and 
although the speed was not particularly entered, it could not be less 
than the rate of two miles and a half an hour. 

*^ As these experiments were fairly made, and by horses of the 
common breed used by farmers, and upon ploughs of various coun- 
Uea, these numbers may be considered as a pretty accurate measure 
of the force actually exerted by horses at plough, and which they are 
able to do without injury for many weeks; but it should be remem- 
bered, that if these horses had been put out of their usual walking 
pace, the result would have been very different. The mean power 
of the draught-horse, deduced from the above-mentioned experi- 
ments, exceeds the calculated power from the highest formula of 



364 



canals: railboaos. 



[pari II. 

Mr. Leslie^ which is as follows : (15 — o)^ = lbs. aToirdnpois for 
the traction of a strong horse, and (12 — v)* ^= lbs. traction of the 
ordinary horse, v = velocity in miles per hour." 



TABLE l. — Shomtig the maximum quantity of labour a Hone of 
averctye strength is capable of performing^ cU different velocities^ on 
Canals^ Railways, and Tumptke-roads. {From Tredgold.) 









Useful eflfect of one horse 


workinir one 




Duration 




day, in tons drawn one mile." I 


Velocity In 


of the day'* 
work at the 






1 


miles per 


traction In 








hour. 


preceding 


lbs. 




On a lerel 






velocity. 




On a canal. 


railway. 


level turn- 
pIke-roML 


milc«. 


houn. 


lbs. 


tons. 


tons. 


tons. 


^ 


u\ 


83i 


520 


115 


14 


3 


8 


83i 


243 


92 


12 


3J 


4 


83i 


153 


82 


10 


4 


83i 


102 


72 


9 


5 


a/o 


83i 


52 


57 


7-2 


6 


2 


83J 


30 


48 


60 


7 


U 


83i 


19 


41 


51 


8 


n 


83^ 


12-8 


36 


4-5 


9 


^?. 


83i 


90 


32 


40 


10 


H 


83i 


6-6 


28-8 


3-6 



Where horse power is employed for the higher velocities, the ani- 
mals ought to be allowed to acquire the speed as gradually as pos- 
sible at the first starting. This simple expedient will save the pro- 
prietors of horses much more than they are aware of; and it deserves 
their attention to consider the best mode of feeding and training 
horses for performing the work with the least injury to their animiu 
powers. 

To compare the preceding table with practice at the higher velo- 
cities, it will be necessary to have the total mass moved, which is 
one-third more than the useful effect in this table. Now, the actual 
rate at which some of the quick coaches travel, is 10 miles an hour; 
the stages average about 9 miles ; and a coach with its load of lug- 
gage and passengers amounts to about 3 tons ; therefore the average 
day's work of 4 coach horses is 27 tons drawn one mile, or 6| tons 
drawn one mile by one horse. The table gives 3*6 tons, addea }- of 
3*6 = 4*8 tons drawn one mile for the extreme quantity of labour 
for a horse at that speed, upon a good level road ; from which should 
be deducted the loss of effect in ascending hills, heavy roads, &C., 
which will make the actual labour performed by a coach-horse ave- 
rage about double the maximum given by the table. The conse- 
quences are well known. 

Accordinjr to Mr. Bevans observations, the horses on the Grand 



CftAP. VII.] TRACTION ON CANALS, RAILROADS. 365 

Jonotion Canal draw 617 tons one mile, at the velocity of 2*45 miles 
per hour. 

According to Mr. TrtdgM^ if V he the maximum velocity of a 
horse, and id any other velocity, the immediate power of a horse is 

250 r f I "- T7 ) ; and, when the weight of the vessel or carriage 

is to the weight of the load, as n: 1, we have V V / =: 

1 H-n 
the effective power ; and d being the hours the horse works in one 

day, the day's work will be V VV in lbs. raised 1 mile, 

1 +n 

and 250 f I ^ -1 J = the force of traction in lbs. But if the force 

14*7 
were immediately applied, the value of V would be —7-); and to 

find the value when the waggons alone are moved, we have 

1 147 14-7 

1 : / , : : --— , : ~" 7~:., — ; = V ; whence the day's work 

v/l + « s/ d y/d (1 + n) ' ^ 

, fl60dv / v^d(\-^n)\ .... . , 

18 I 1 -—\-)-= ' ] I which IS a maximum when 

1 + n \ 14-7 /' 

06 

= d. Consequently, when the velocity is given, we 



r* (1 + «) 



06 

bftve -77- r equal the duration of the day's work in hours: 

tr(l -f 11) ^ 

8000 / 0*8 \ 

and — rr Tj- = the effective day's work; and 250 f I — -— j = 

83^ lbs. But we may assume n to be always so near ^, as not to 

affect the result : and then, —- = rf, and = the day's work in 

2 
lbs. or very nearly - tons raised one mile. This, being combined 

with the Dumbers of the preceding table, gives the effect of a horse 
on canals, railroads, and turnpike-roads. 

It must, however, be here added, that although the deductions 
from Mr. Tredgdd^% valuable tables, as to the effects on canak^ are 
tolerably accurate up to rates of 4 or 5 miles per hour, yet, when 
boats are moved on canals at rates of from 9 to 12 or 14 miles 
per hour, the circumstances of the resistances undergo an essential 
ehaoge. The resistance, in fact, becomes so small, that passage- 



366 TaACTION ON CANALS, RAILBOADS. [PART II. 

boats DOW trayel at these high velocities; and it is hence probable 
that railroads and canals will admit of a competition such as the sup- 
porters of railroads never anticipated. 

I shall here briefly detail some of the facts, as they have been 
given in a letter widely circulated by Mr. W. Grakame of Glasgow, 
in the Nautical Magaziney and other places. 

From the traffic by canal boats, which has been actually going on 
during the last two years and a half, on the Paisley canal, we learn 
this remarkable fact, that, while a speed of ten miles per hour has 
been maintained by the canal boats, the banks have sustained no in- 
jury whatever. The cause of injury, in truth, has been entirely sup- 
pressed by the velocity of the boat, which passes along the water 
without raising a ripple. 

About two years ago, measures were adopted for increasing the 
speed of the boats on the Paisley or Ardrossan canal. This canal is 
by no means favourable to such experiments, being both serpentine 
in its course, and narrow : it connects the town of Paisley with the 
city of Glasgow, and the village of Johnstone ; the distance being 
about twelve miles. The boats employed on this canal are 70 feet 
in length, and 5*6 broad, and carry, if necessary, upwards of 120 
passengers. They are formed of light iron plates, and ribs covered 
with wood, and light oiled cloth, at a whole cost of about 1 25/. They 
perform stages oJT four miles in an interval of time varying from 22 
to 25 minutes, including all stoppages, and the horses run three or 
four of these stages alternately every day. The passengers are under 
cover, or not, as they please, no difference being made in this parti- 
cular ; and the fare is one penny per mile in the first, and three 
farthings per mile in the second cabin. 

The horses drawing the canal boat are guided by a boy, who rides 
one of them ; and, in passing under bridges at night, a light is shown 
in the bow of the boat, by which he sees his way, and which light 
is closed when the bridge is passed. Intermediate passengers are 
also accommodated to distauces even as small as a mile ; and the faci- 
lity with which the boat stops, when reHeved from the drawing force, 
is such as avoids all danger whatever. The expense of conveying a 
load of eight tons at a rate of nine or ten miles per hour^ including 
all outlay, interest, and replacement of capital, is not more than i\d. 
per mile. It is also ascertained that one ton weight may be carried 
on a canal at nearly the same speed as on the railway, at about 1^ 
per mile, including an allowance for interest and replacement of 
capital. 

It is also believed, that if the breadth and curvature of the Paisley 
canal admitted boats of 90 feet length, instead of 70, they would 
carry more passengers by one-half without an additional expense, and 
a decrease of labour to the horses. 

The foregoing has been deduced from calculations founded oo 
the observation of facts relating to the wear and tear of boats and 
horses, and the absolute resistance which these boats meet with in 
passing through the water. On this subject it has been observed. 



CHAP. VU.] TRACTION ON CANALS, RAILROADS. 3^7 

that, in addition to the common resistance of the water to the 
motion of the boat, a wave, or body of water, is also raised before 
it, varying in its height according to the velocity of the boat, and 
constantly presenting an obstacle to ber progress, providing that she 
only moves through the water at a certain slow rate. The height of 
this wave will then amount to nearly two feet, often overflowing the 
banks of the canal, and, from the obstruction it occasions, eventually 
obliging the boat to be stopped. Now, if, instead of stopping the 
boat when this wave is raised, her velocity be increased beyond what 
it had then been, she advances and passes over it, and leaves it to 
subside in her wtike, which it does, and the water becomes perfectly 
still. The same horses, drawing the boat at this increased speed, are 
found to perform their work better, the resistance to their progress 
having become less ; and the more the velocity of the boat is thus 
increased, the less resistance she meets with, merely having to cut 
the still water instead of the wave. It is a curious fact, that the 
wave produced by the approach of a slow canal boat is often observed 
at the distance of a mile, and upwards, along the canal, before the 
arrival there of the boat. But, in the case of the high wave being 
raised by the Paisley canal boat, it is customary to stop the boat, and 
after it has subsided to start again at a greater velocity. When the 
boat 18 to be stopped for any purpose, as her speed decreases the 
wave rises in proportion, and washes over the banks, until the motion 
of the boat becomes so small Iks to produce none. The discovery is 
doahtless a very important one, and, if turned to account, is likely to 
produce a material alteration in the rate of transport on canals. It 
was not known until these experiments were made, that if a boat, 
from a state of rest, was dragged along a canal, in proportion as her 
•peed increased to a certain limit, that the power required was 
greeier: bat that, if she were started at, and preserved a speed ex- 
ceeding the same limit, the power required would be less, and would 
decrease as her velocity increased, in fact, from a certain velocity 
there seems to be no limit to the rate at which a boat, as far as ani- 
mal power can be applied, may thus pass through the water ; and as 
the rate increases the power required decreases. On this principle 
it is that the boats on the Paisley canal, with ninety passengers in 
them^ are drawn by horses at a speed of ten miles an hour; while it 
would kill them to draw the same boat along the canal at six mile$ 
an hour. A boat might indeed travel fifteen or twenty miles an 
hour eancr than at six miles. The former of these velocities has 
already been attained by Mr. Grahamey along a distance of two miles, 
and is considered by him safer both for the boat and tlie canal. 

As a proof of what may be done by this method of carriage, Mr. 
Chrakame states that he has performed a voyage of fifty-six miles 
along two canals in six hours and thirty-eight minutes, which in- 
cluded the descent of ^ye^ and the ascent of eleven locks, the pas- 
sage of eighteen drawbridges where the tracking-lino was thrown 
off, and sixty common bridges, besides a tunnel half a mile long ; all 
of course prodacing some delay. The boat which performed this 



368 TRACTION UN CANALS AND ROADS. [PART II. 

was sixty-nine feet long, and nine broad, drawn by two borses, and 
carried thirty-three passengers, with their luggage and attendants. 

These facts furnish great encouragement to canal companies, to 
improve the construction of their boats and the speed on their canals ; 
and thus, probably, in some situations, supersede the necessity of 
railroads. 

Mr. Macneill, the assistant engineer upon the Holyhead road, under 
Mr. Telford, in the course of his examination before a committee of 
the House of Commons, on steam-carriages, railroads, &c. gave the 
subjoined curious information. 

Well-made roads, formed of clean hard broken stone, placed on a 
solid foundation, are little affected by changes of atmosphere ; but 
weak roads, or such as are imperfectly formed i^dth gravel, flint, or 
round pebbles, without a bottoming of stone, pavement, or concrete, 
are much affected. 

On the generality of roads, the proportional injury from the wea- 
ther and traffic is nearly as follows. When travelled by fast coaches : 
from atmospheric changes 20 ; coach-wheels 20 ; horses' feet 60 = 
1 00. When travelled by waggons : atmospheric changes 20 ; waggon- 
wheels 35'5 ; horses' feet 44*5 = 100, Has ascertained, from a num- 
ber of observations, that the wear of the iron tire of fast-going coach- 
wheels is, compared with that of the shoes of the horses which draw 
them, as 326*8 to 1000, or as 1 to 3-4ths nearly ; and infers that the 
comparative injury done by them to roads is nearly in the same pro- 
portion. In the case of slow-going carriages and horses the propor- 
tion is as 309 to 360, or as 1 to 1*16, or nearly 1 to 1|. The tire of 
the wheels of the fast-going coaches last from two to three months, 
according to the weather, the workmanship, and quality of iron; 
about 20 years ago, it did not lost seven days on an average. Coach- 
horse shoes remain in use about thirty doys; waggon-horse shoes 
about five weeks on an average. Where roads are weak, and yield 
under pressure, the injury caused by heavy wheels is far greater than 
on solid firm roads. 

It was found, in one instance, that the wear of haid stone, placed 
on a wet clay bottom, was four inches, while it was not more 
than half an inch when placed on a solid dry foundation. On the 
Highgate archway road, the annual wear is not more than half an 
inch in depth. To the same gentleman we owe the following useful 
table. 

TABLE II. — Tlie general Result of Experiments made with a Stage 
Coachy weighing^ exdtuive of seven passengers^ 18 ctct.^ on the same 
piece of road^ on different inclinations^ and at different rtUes of 
velocitgy furnish the following statement. 

Rate of Inclination. Rates of Travelling. Force required. 

I in 20 6 miles per hour 268 lbs. 

1 in 26 6 213 

1 in 30 6 165 



CHAX». 



VII.] 

Hate of Inclination. 
1 



TRACTION ON ROADS. 



369 



Rates of Travelling. Force required. 

n 40 6 miles per hour 160 lbs. 

In 600 6 Ill 



20 
26 
30 
40 



in 600 8 



20 
26 
30 
40 



10 
10 
10 
10 



n600 10 



296 
219 
196 
166 
120 

318 
225 
200 
172 
128 



B B 



370 STBEKOTH OF MATERIALS. [PART II- 



CHAP. VIII. 

STRENGTH OP MATERIALS. 

Sect. I. EestdU of Experiments^ and Principles upon which tkjf 
should he practically applied. 

By most writers on the strength of materials, it has been customarj 
to start with the assumption of the three under-mentioned principles, 
and upon them to construct a theory by means of which they afte^ 
wards deduced from experiments on rectangular bars tlie strength of 
beams of the same material, but of various forms of transTerse sec- 
tion. Upon, however, testing these deductions by experiment, thcj 
are found in many cases to be entirely erroneous, and to such an 
extent as to render their practical application not only uncertain, bat 
frequently highly dangerous. Such being the result of experience, 
it becomes important to examine these principles more closely, in 
order to discover wherein the cause of this erroneous result consists. 

The principles which we allude to are (in the language of Tred- 
gold*) as follows: — 

" The first is, that the strength of a bar or rod to resist a given 
strain, when drawn in the direction of its length, is directly pro- 
portional to the area of its cross section ; tchile its Mastic pc^n^ 
remains perfect^ and the direction of the force coincides wiUi the 
axis. 

" The second is, that the extension of a bar or rod by a fore* 
acting in the direction of its length is directly proporUonal to tbe 
straining force, when the area of the section is the same ; whiU ^ 
strain does not exceed the elastic power. 

" The third is, that while the force is within the elastic power of^ 
material^ bodies resist extension and compression with equal forces. 

It should be remarked, that each of the foregoing propositions i« 
only asserted to hold good " while the strain does not exceed the 
elastic power of the material." By which is meant that tbe force 
applied shall not be sufficient to cause any permanent displacement of 
the particles, and that when the straining force is removed, tbe body 
will not have acquired any set, but will reassume its former sbspc 

* Practical Treatise on the strength of cast iron. Fourth edition, pw ISi 



STRSNOTH OF MATBKIALS. 371 

». Tredgold and most other writers upoD this subject 
d that no permanent displacement or set takes place 
ining force amounts to about one-third of tho ultimate 
be material, and have consequently assumed that these 
ght be applied without error to determine the strength 
sv of materia], of any form, and subject to any kind 

more careful experiments have, however, shown conclu- 
permanent displacement or set takes place with a much 
I than one-third of the ultimate strength, and that in 
able set is produced by a straining force very much less 

which the material will be likely to be exposed in 
cfa being the case, it is obvious that these principles (as 
»Te) are inapplicable to any practical case. As, how- 
ire been disposed to imagine that such a strict limitation 
icmtion is not necessary, and that they ])ractically hold 
ben the force applied eaxeeds the elastic power of the 
ill be desirable to show that not only is such a supposi- 
nroneous, but that the principles themselves, even with 
t, so far from being demonstrable, are, on the contrary, 
loubtful. 

■ertion, that the strength of a rod or bar subjected to 
n is directly proportional to the area of its cross section, 
irat sight it may appear obvious enough, nevertheless is 
practice to be strictly correct ; for, if the material is of 

it found that the strength of the exterior hard crust is 
bat of the interior substance, and therefore that the 
bicb the perimeter of the section bears to its area is a 
requiring to be taken into consideration ; not to men- 
urangement of the particles during the cooling of the 
ipon which arrangement the strength of the substance so 
lends, is greatly influenced by the external form of its 
i on the other hand, in the case of a fibrous material, aa 
da or timber, in consequence of the fibres not laying 
illel to each other, in cutting the bar to the required I 

may of the exterior fibres will be cut transversely, and > 

rfbre be capable of affording so great a proportionate li 

m similar fibres within the more central portion of the jj 

ieh it follows, that the tensile strength of a bar of any 

i directly proportional to its sectional area, although tho I 

» trifling as not to require attention in most of the cases ij 

i tfaemselyes in practice. { 

I law, that the extension of a bar or rod exposed to a \^ 

m directly proportional to that strain 90 long cu ike area \ 

remains unaltered^ is from the very circumstances of \ 

■ible, since it is not capable for a bar to become elon- 
biTing its transverse section proportionately diminished, 

we cannot conceive the bar in two successive degrees I 

l^Ting in both the same sectional area. And although 

B B 2 



372 STRENGTH OF MATBBIALS. [PABT IL 

the correct law subsisting between the elongation and the force pro- 
ducing it, has not yet been satisfactorily determined, there is eierr 
reason to believe that they are not directly proponionil to em 
other. 

With regard to the third law, which may be more definitely ititfld 
as follows, viz. : — that within the elastic power of the materaJ,bo&i 
require as great a force to alter their dimensions to any giTen exttnt 
by compression as by extension ; we may remark, that an? propeilf 
depending upon the straining force being within the elastie power tf 
the material is incapable of any practical application, and tbeRfim 
that this law, as limited above, even assuming it to be correet, doM 
not require to be included in any practical inquiry into the kvt 
which regulate the strength of materials. And without soch liniti^ 
tion the law is at utter variance with the results of experiment, net 
no fact is now better ascertained than that cast iron requires a dmIi 
greater force to compress it to any given extent, than to extend it to 
the same extent. 

Such then being the result of our investigation into the socoiteeB 
of these principles with practical experience, we cao no longer It 
surprised that the deductions drawn from them by Tredgc^ mI 
others, should be equally at variance with the same experieaefc 
Indeed the errors into which they have fallen appear to hate beet 
caused by reasoning upon hypothetical principles — attempthig to foia 
a complete theory of the strength of materials upon data neitlMr 
sufficient in quantity nor certainty. And even now, although una 
has been done, since the time when Tredgold wrote, by HodgkiMi 
Fairbaim, Thomas Cubitt, Barlow, and some others, this inraffieienef 
has not yet been supplied, and we are still in ignorance upon OMJ 
points, without a knowledge of which it would be impossible to fi»« 
any complete theory of the strength of materials aafiicient in Hi 
extent for all practical purposes, and of whose accuracy we cooM he 
so far sure as to allow of its application without fear of erroseov 
results. 

In order to show in what points the deficiency in oor knowle^ 
upon this subject consists, and to explain the general prindplei vp** 
which the strength of materials actually depends, we will hiiiif 
investigate the subject 

If A BCD, fig. 230, be a rectangular beam, subjected to fcj 
pressures Pj, P.^, and P.,, the two former of which arc eqoal to eidi 
other, and to half P,, and if Pi and P, be applied at the two eAUHJi 
ties of the beam, and P.< at its center in an opposite direetioBi ■• 
beam will be deflected, the upper surface A C becoming cooviX» •* 
the lower surface BD concave. Then, if the material compoiiogvt 
beam be supposed capable both of extension and com prcMi o^ w* 
fibres on the convex side will be extended, and those on the con** 
side compressed ; there will, however, be a certain curved aupeifc* 
£ F within the beam (called the neutral superficies), the fibres tfti^ 
in which will not be either extended or compressed. 

Now, if tangents EG, FU, to this curved neutral miperficiei it ^ 



CHAP. VIII.] STRENGTH OP MATEBIAL8. 373 

extremities of the beam be drawn, and from them perpendiculars K L, 
M N, pasnng through the point I, in which the neutral superficies inter- 
lects the plane of rupture OQ; then will the triangle NIL be equal to 
the sum of the extensions of the whole length of all the extended fibres; 
tnd the similar triangle KIM will be equal to the sum of the com- 
pressions of the whole length of all the compressed fibres; and if any 
•operfides be taken between AC or BD and the neutral superficies, the 
portions of such superficies, as t^ i,, yi^s? contained between KL, and 
II N, will be equal to the extension or compression of the whole length 
of the fibres situate in that superficies : from whence it follows that the 
amount of extension or compression of any fibre is directly propor- 
tional to its distance from I, the neutral axis. 

If now any number of points be taken in the depth of the beam, 
as ij, f^ 1^ yj, ^29 y»* ^^' (%• 231), and lines be drawn through 
them perpendicular to O Q, each equal in length to the force required 
to extend or compress the fibres to the extent to which such fibres 
are actually extended or compressed, and which extent will be propor- 
tiona] to their distances from the neutral axis, and a curved superficies 
^^i^iCtdid^e^l^bj^a^ be drawn through the ends of all these lines, 
then will the solid contained between the plane of rupture O Oj Q^ Q 
and the curved superficies, be equal to the resistance of all the extended 
and compressed fibres, or to the transverse strength of the beam. 

Now, if £ equals the solidity of ai^^lil^^^^a) ^^ ^ equals the 
soKdifcy of ^iCid^^d^c^I^ also ^and A equal the perpendicular distances 
of their center of gravity from the neutral superficies, and ^L the dis- 
tance of the points where the pressures P^ and F^ are applied, from the 
neutral axis ; we have from the principle of the equality of moments, 

iLP^ = E^ + CA; 

or, ainoe El must be equal to C^, and P^ equals \ P,, we have 

5l'P,= E» (I.) 

■ad this formula will apply» whatever form the transverse section of 
the beam may be, whether rectangular as fig, 281, cylindrical as 
fig. 232, or X shaped as fig. 233. 

It is further evident £at, when the beam is upon the point of 
breaking, the upper fibres at O must be strained to the utmost extent 
Id whkh they can be, before yielding, and that as this extent does 
not depend upon the form of the beam, but only upon the elastic 
properties of tne material, for the satee material the length a^ O will 
renuitn constant in all cases. From this it follows that the lines 
■i, f^ •„ 7„ y^ 73, &c. (fig. 231), which are the ordinates of the 
enire a^hA^e^di^ will also be constant, but that their distances from 
E, or the Miacisse of the curve will vary directly as the distance O I 
of the neutral axis from the upper surface of the beam. 

In rectangular beams, or others, whose transverse sections are 
■mikr figures, the distance O I, and therefore the area of the curve 



374 STRBNOTH OP MATERIALS. [PART II. 

O a^ b^ Ip will yary as the depth of the beam, and the solidity of the figure 
Oa^b^ IjI^^^.a^O will vary as the depth and the breadth, or Eo: bd; 
also, since the forms of the beams are similar^ the distance i of the 
center of gravity of the solid E from the neutral superficies, mnst vary 
as the depth of the beam ; and therefore, in rectangular beams, the 
ultimate strength of the beam varies directly as the square of the depth 
multiplied by the breadth, and inversely as the length ; or, putting 
S (= Pj H- Pj rs P.^) for the breaking weight applied in the center,^ 
we have 

Sac^* (II.) 

or, when the transverse sections are similar figures^ </^ being any 
similar dimensions in both, 

d^ 
Sa^ (III.) 

From the preceding investigation it is evident that we cannot 
deduce any general formulse by which to determine the transverse 
strength of a beam of any given form of section, without knowing 
the position of the neutral axis, the limit to which the fibres may 
be extended before yielding, and the law which subsists between the 
extension and compression of the material and the force required to 
produce such extension and compression ; upon none of which sub- 
jects have we any certain knowledge. 



Sect. IL Stren^ ofMateriaU to resist tensile and crushing strains* 
Strength of Columns. 

The first principle alluded to in the foregoing section, via., that the 
strength of a bar or rod to resist a tensile strain in the direction of 
its axis, IS proportional to the area of its cross section, is not, for the 
reasons there mentioned, strictly correct. And although perhaps for 
ordinary practical purposes, or where the sections to be compared aie 
large, no error of importance wonld result from the adoption of this 
principle, it is very necessary, in experiments whose object is tbe 
determination of the actual strength of any given materia], that doe 
regard should be had to all circumstances, (however trivial they msj 
seem,) which may in any way affect the accuracy of the results. 

The only late experiments upon the tensile strength of cast iron, 
now before the public, are those of Mr. Eaton Hodgkinson, first 
published in a paper communicated to the British Association, and 
subsequently in his '^ Experimental Researches." The mean tensile 
strength of a square inch of cast iron, as deduced from these experi- 
ments, is only 10,560 lbs., or 7 tons 7*85 cwt.; the lowest heing 
12,993 lbs., and the Iiighest 21,907 lbs. In a series of experiments, 



OHAP. VIII.] C0HE8IVB 8TRINOTH OF CAST IRON. 375 

howeTer, which have been tried by Mr. Thomas Oubitt*, during a 
period of several years, we find the strength of cast iron to be con- 
siderably higher than Mr. Hodgkinson's estimate: the average of 
twenty-five specimens, of which eight were unsound, gave 27,140 lbs. 
for the tensile strength per square inch section ; and, exchiding the 
unsound pieces, the average of the others gave 27,773 lbs., or 12 tons 
8 cwt. The highest broke with 32,997 lbs. per square inch, and the 
lowest^ even of the unsound pieces, broke with 21,471 lbs., or only 
a little under Mr. Hodgkinson's best speeimen. 

The anomaly between these results is very remarkable, since in 
both cases the experiments were conducted with the greatest care ; 
we think, however, that Mr. Cubitt's results are entitled to the pre- 
ference, for this reason, that while any cause might easily operate to 
lessen ihe strength of the iron, we cannot conceive it possible that 
any want of care in the experiments or bad arrangement of the 
apparatus could cause the iron to bear more than it otherwise would 
have done, providing that the means of measuring the force exerted 
in producing fracture were correct ; and these, we do not hesitate to 
aay, were not open to any exceptions. It is certainly difficult to 
account for so wide a difference in the results of these experiments, 
but we cannot but think that the form chosen by Mr. Hodgkinson, 
for the transverse section of the bars upon which he experimented, is 
open to objection. The form adopted by him is shown in fig. 234, 
being that of a cross ; and the reason which induced him to choose 
this form, was " to obviate the objection made by Mr. Tredgold 
(Art. 79 and 80,) and others, against the conclusions of former 
experimenters." The objection of Mr. Tredgold here alluded to was, 
that if the strain did not pass through the center of gravity of the 
section in the place where the rod broke, the different parts of that 
section would be exposed to a different strain, and therefore,' that the 
results obtained would be lower than the truth. We quite agree with 
Mr. Tredffold upon the importance of this objection, but we cannot 
bat consider, that the form chosen by Mr. Hodgkinson is more open 
to this objection than the square or cylindrical form would have been. 
For, whatever might be the form of section of the rod, the same means 
of fixing the ends of it might be used, and since it depends entirely 
upon the accuracy with which that is done whether the axis of the 
rod coincides with the straight line between the points of attachment, 
(which is the direction of the straining force), we may suppose that 
the lateral distance between these two lines at the breaking section 
would be the same, whatever was the form of that section; such 

* I am much indebted to Mr. Thomas Cubitt, for the liberal manner in 
wliich he has placed at my disposal the results of his very careful and ela- 
bormte experiments upon this subject. Ui« experiments on the set, deflection, 
and ultimate strength of cast iron girders, extend over a period of many years, 
and p o i Bs eis the immense practical advantage of having been made upon large 
girders. These experiments, amounting in number to upwards of 2000, have 
been taken under the direction of Mr. Dines; and, having witne98ed some of 
them myself, I can bear testimony to the care which has been taken to ensure 
aecurmte results, and to remove all imaginable causes of error.— H. L. 



370 C0HE8IVB 8TRBN0TH OP MATERIALS. [PART II. 

being the case then, we hold that the strength of the cnicifonn sec- 
tion would be more affected than the square, since in the former the 
metal is situated further from what would then be the neutral axis 
of the section, and exposed therefore to greater differences of strain. 
This form is also open to the practical objection that it is very diffi- 
cult to obtain a perfectly sound casting from it ; it asually happens 
that the iron is not perfect at the point of intersection, (a, fig. 234,) 
and also to our remarks at p. 371, that the proportion of the peri- 
phery to the area may influence the strength. 

Mr. Cubitt has also tried a few experiments upon wrought iron, 
which gave for the average tensile strength per square inch, 
58,952 lbs., or 26 tons 6*3 cwt. 

The metals differ more widely from each other, in their elastic 
force and cohesive strength, than the several species of wood or vege- 
table fibres. Thus, the cohesion of fine steel is about 135,000 lbs. 
for the square inch, while that of cast lead amounts only to about the 
hundred and thirtieth part, or 1800 lbs. 

According to the experiments of Mr. George Rennie, in 1817, the 
cohesive power of a rod an inch square of different metals, in pounds 
avoirdupois, with the corresponding length of the modulus of elasticity 
in feet, is as follows : — 

Pounds. Feet. 

Cast steel 134,256 ... 39,455 

Swedish malleable iron ... 72,064 ... 19,740 

English ditto 55,872 ... 16,938 

Cast iron 19,096 ... 6,110 

Cast copper 19,072 ... 5,003 

Yellow brass 17,958 ... 5,180 

Cast tin 4,736 ... 1,496 

•Cast lead 1,824 ... 348 

It thus appears, as Professor Leslie remarks, that a vertical rod o£ 
lead 348 feet long would be rent asunder by its own weight. Ther* 
best steel has nearly twice the strength of English soft iron, and this 
again is about three times stronger than cast iron. Copper and brass 
have almost the same cohesion as cast iron. This tenacity is some— 
times considerably augmented by hammering or wire-drawing, that= 
of copper being thus nearly doubled, and that of lead, according t o^^ 
Eytelwein, more than quadrupled. The consolidation is produced ^ 
chiefly at the surface, and hence a slight notch with a file will 
materially weaken a hard metallic rod. 

Professor Leslie has given the following tabular view of the 
lute cohesion of the principal kinds of timber, or the load which^ 
would rend a prism of an inch square of each ; and the altitude of tb^? 
prism which would be severed by the action of its own weight. 

Pounds. Feet. 

Teak 12,915 ... 36,049 

Oak 11,880 ... d2,900 

Sycamore 9,630 ... 35,800 

Beech 12,225 ... 38,940 



CHAP. VIII.] STRENGTH OF CORDAGE. 377 

Pounds. Feet. 

Ash 14,130 ... 42,080 

Elm 9,720 ... 39,050 

Memelfir 9,540 ... 40,500 

Christiana deal 12,34G ... 55,500 

Larch 12.240 ... 42,160 

The following is the result of Professor Barlow's experiments upon 
the cohesive strength of various kinds of timber. 

Pounds 
per square inch. 

Fir 12,203 

Ash 17,077 

Beech 11,467 

Oak 10,389 

Teak 15.090 

Box 19,891 

Pear 9,822 

Mahogany 8,041 

The cohesion of hempen fibres is, for every square inch of their 
transverse section, 6400 Ihs. The best mode of estimating the strength 
of a rope of hemp is to multiply by 200 the square of its number of 
inches in girth, and the product will express in pounds the practical 
strain it may be safely loaded with ; for cables, multiply by 120 instead 
of 200. The ultimate strain is probablv double this, as will appear 
from the account following of Du Hamei s experiments. If yams of 
180 yards long be worked up into a rope of only 120 yards, it will 
lose one-fourUi of its strength, the exterior fibres alone resisting 
the greatest part of the strain. The register cordage of the late 
Captain Huddart exerts nearly the whole force of the strands, since 
they suffer a contraction of only the eighth part in the process of 
combining. 

For the utmost strength that a rope will bear before it breaks, a 
good estimate will be found by taking one-fi/lh of the square of the 
girth of the rope, to express the tons it will carrv. This is about double 
the role for practice which we have given above ; and is, even for 
an ulterior measure, too great for tarred cordage, which is always 
weaker than white. 

The following experiments were made by Mons. Du Hamel, at 
Rochfort, on cordage of three inches (French) circumference, made 
of the best Riga hemp, August 8th, 1741. 

White. Tarred. 

Broke with a strain of 4500 pounds «.. 3400 pounds. 

4000 „ ... 3300 „ 

4800 „ ... 3258 „ 

August 25th, 1743. 

4600 „ ... 3500 „ 

5000 „ ... 3400 „ 

5000 „ ... 3400 „ 



378 snunroTH op oobdaoi. [part ii. 

September 23, 1746. 
White. Tmrred. 

Broke with a itruo of 3880 pounds ... 3000 pounds. 
4000 „ ... 2700 „ 
4200 „ ... 2800 „ 

A parcel of wbite and tarred cordage was taken out of a quantity 
wLich had been made February 12, 1746. 

It was laid up in the Magazine, and comparisons were made from 
time to time, as follows : — 

Wbite. Tarred. Difference. 

1746, April 1 4th, 2645 pounds 2312 pounds 333 pounds. 

1747, Mav 18th, 2762 „ 2155 „ 607 

1747, October 21st, 2710 „ 2050 „ 660 „ 

1748, June 19th, 2575 „ 1752 „ 823 „ 

1748, October 2nd, 2425 „ 1837 „ 588 „ 

1749, Sept. 25th, 2917 „ 1865 „ 1052 „ 

M. Du Hamel says, that it is dcnnded by experience, that white 
cordage in continued service is one-third more durable than tarred ; 
secondly, it retains its force much longer while kept in store ; thirdly, 
it resists the ordinary injuries of the weather one- fourth longer. 
These obsenrations deserre the attention of pmctical men. 

Mr. B, Becan has favoured the author with a tabular view of his 
results with regard to the modulus of cohesion^ or the length in feet 
of any prismatic substance required to break its cohesion, or tear* 
it asunder. 

Bet>an% Remits, 

Feet. 

Tanned cow's skin 10,250 

Tanned calf skin 5,050 

Tanned horse skin 7,000 

Tanned cordovan 3,720 

Tanned sheep skin 5,600 

Un tanned horse skin 8,900 

Old harness of thirty years 5,000 

Hempen twine 75,000 

Catgut, some years old 23,000 

Garden matting 27,000 

Writing-paper, foolscap 8,000 

Brown wrapping-paper, thin 6,700 

Bent grass, (holcus) 79,000 

Whalebone 14,000 

Bricks, (Fenny Stratford) 970 

Bricks, (Leighton) 144 

Ice 300 

Leicestershire slate 7,300 



CHAP. VIII.] 



8TRBN0TH OF COLUMNS. 



879 



STRENGTH OF COLUMNS. 

Upon the subject of the strength of columns, we are entirely 
indebted to Mr. Hodgkinson, who, after a very careful and accurate 
series of experiments, has deduced formulee of immense practical 
value, and has clearly shown the principles upon which the strength 
of materials, submit to a compressing force, depends. 

The results of his experiments are briefly as follows : — That when 
weights are applied to prisms, or cylinders, the shorter ones usually 
bore more than the longer ones, and that the strength may be con- 
sidered to vary as the area. That, when the diameter or least 
lateral dimension is less than the height of the prism, fracture takes 
place, by the separation of either a pyramid, cone, or wedee, 
(depending upon the form of the prism,) the angle of whose sides 
is constant for the same material, and about 34° for cast iron. And 
further, that the mode of fracture and the ultimate strength of a 
prism varies but slightly with a variation in its height, so long as it is 
not less than its diameter or least lateral dimension, or greater than 
four or ^ye times the same dimension; when higher, it begins to 
bend, and its strength decreases. 

Mr. Hodgkinson finds that the strength of a column is very much 
influenced by the manner in which the ends arc fixed ; when both 
ends are rounded, so that the column may turn upon them as a 
center, its strength is only one-third of that of another column of 
precisely similar dimensions, but whose ends are flat and incapable 
of motion ; and that, if one end is flat and immovable while the other 
is rounded, the strength is two-thirds of what it would have been had 
both ends been flat. 

The following table exhibits the results of his experiments on solid 
and hollow cylindrical columns of cast iron, in which S is the break- 
ing weight in tons ; D the external diameter, d the internal diameter, 
both in inches ; and / the length or height of the column in feet. 



Kind of Cdumn. 



With both ends round,! With both endii flat. 

when the height of the'when the height of the 

column is not less than'oolumn is not less than 

15 times iu diameter. 30 times its diameter. 



Solid cylindrical cast ironl 
columns j 

Hollow ditto ditto 



S = 14-9 
8 = 13 



p3^ 



/1-7 



8 = 44-16 
8 = 44-3 






/1-7 



When the height of the column is less than that mentioned in this 
table, namely, fifteen times its diameter for columns with both ends 
rounded, and thirty times the same with both ends flat, the strength 
of the column becomes modified in consequence of its being then 
partially crushed as well as bent. If C equal the force which would 
be required to erush the column without flexure, 8 the strength as 



380 8TRBN0TH OF COLUMNS. [PART II. 

calculated by the above formulse for long columns, and S^ the actual 
strength, we have 

'■-JTT' <>^-' 

The value of c, or the weight required to crush a square inch of 
cast iron, as deduced from his experiments, is 1 07,750 lbs., or 48 tons 
2 cwt. ; therefore, the crushing force is to the tensile, according to 
Mr. Hodgkinson, as G'507 is to 1 ; or, taking Mr. Hodgkinson's value 
for the crushing strength, and Mr. Cubitt's for the tensile, the ratio 
becomes as 3*88 is to 1. 

The strength of a column of cast iron of given dimensions being 
1000, the strength of a column of wrought iron of the same dimen- 
sions would be 1745, of cast steel 2518, of Dantzic oak 108*8, and 
of red deal 78-5. 

The weights required to crush cubes of the quarter of an inch of 
certain metals, according to the experiments of Mr. Rennie, arc as 
follows : — 

Pounds. 

Iron cast vertically 11,140 

Iron cast horizontally 10,110 

Cast copper 7,318 

Cast tin 966 

Cast lead 483 

Cubes of an inch are crushed by the weights annexed : — 

Pounds. 

Elm 1,284 

White deal 1,928 

English oak 3,860 

Craigleith freestone 8,688 

Cubes of an inch and a half, and consequently presenting a section 
of two and a quarter times greater than the former, might be expected 
to resist compression in that ratio. They are crushed, however, with 
loads considerably less. 

Pounds. 

Red brick 1,817 

Yellow baked brick 2,254 

Fire brick 3,864 

Craigleith stone, direction of the strata... 15,560 

Ditto across the strata 12,346 

White statuary marble 13,632 

White-veined Italian marble 21 ,783 

Purbeck limestone 20,610 

Cornish granite ..« 14,302 

Peterhead granite 18,636 

Aberdeen blue granite 24,536 

These facts show the comparative firmness of different materials ; 
but it is to be regretted that such results are not of much praetical 
value, since they are confined to a very narrow scale, and applicable 



CHAP. VIII.] MODULUS OP ELASTICITY. 381 

only to cubical blocks. While the breadth remains the same, the 
resistance appears to depend on some unascertained ratio of the 
altitude of the column. 



Sect. III. EUutieiiy and ElongcUion of Bodies suhfected to a 
crushing or tensile strain. 

The modtdtis of the elasticity of any substance is a column of the 
same substance, capable of producing a pressure on its base which is 
to the weight causing a certain degree of compression, as the length 
of the substance is to the diminution of its length. 

The modulus of elasticity is the measure of the elastic force of 
any substance. 

A practical notion of the moduius of elasticity may be readily 
obtained. Let i be the quantity a bar of wood, iron, or other sub- 
stance, an inch square and a foot in length would be extended or 
diminished by the force f; and let / be any other length of a bar of 
equal base and like substance ; then 

1 : / : : I : ^, or /f = ^, 

where A equals the extension or diminution in the length /. 

The modulus of elasticity is found by this analogy : as the dimi- 
nution of the length of any substance is to its length, so is the 
force that produced that diminution to the modulus of elasticity. Or, 
denoting the weight of the modulus in pounds for a base of an inch 
square by m, it is 

I :/: : 1 : w = '^. 

E 

And if v be the weight of a bar of the substance one inch square 
and one foot in length ; then, if M be the height of the modulus of 
elasticity in feet, we have 

M=-^ (V.) 

The weight of the modulus of the elasticity of a column being 
m, a weight bending it in any manner f the distance of the line 
of its application from any point of the axis D, and the depth of the 

column, dy the radius of curvature will be . 

The distance of the point of greatest curvature of a prismatic 
beam, from the line of direction of the force, is twice the versed sine 
of that arc of the circle of greatest curvature, of which the extremity 
is parallel to that of the beam. 

When the force is longitudinal, and the curvature inconsiderable, 
the form coincides with the harmonic curve, the curvature being pro- 
portional to the distance from the axis ; and the distance of the point 



382 MODULUS OF ELASTICITY. [PABT II. 

of indifference from the axis becomes the secant of an arc propor- 
tional to the distance from the middle of the column. 

If a beam is naturally of the form which a prismatic beam would 
acquire, if it were slightly bent by a longitudinal force, calling its 
depth dy its length /, the circumference of a circle of which the 
diameter is unity c, the weight of the modulus of elasticity m, the 
natural deviation from the rectilinear form A, and a force applied at 
the extremity of the axis /y the total deviation firom the rectilinear 
form will be 

^^VA^»_ 

It appears from this formula, that when the other quantities 
remain unaltered, t! varies in proportion to A, and if A = O, the 
beam cannot be retained in a state of inflection, while the denominator 
of the fraction remains a finite quantity; but when d^trm = 12 V f^ 
A^ becomes infinite, whatever may be the magnitude of A, and the 
force virill overpower the beam, or will at least cause it to bend so 
much as to derange the operation of the forces concerned. In this 

-^ j . -— , -8225 — w, which is the force capable of 

holding the beam in equilibrium in any inconsiderable degree of 
curvature. Hence, the modulus being known for any substance, we 
may determine at once the weight which a given bar nearly straight 
Is capable of supporting. For instance, in fir wood, supposing its 
height 10,000,000 feet, a bar an inch square and ten feet long may 
begin to bend with the weight of a bar of the same thickness, equal 

in length to -8225 x -— :: tt^ x 10,000,000 feet, or 671 feet; 

120 X 120 

that is, with a weight of about 120 lbs. ; neglecting the effect of the 
weight of the bar itself. In the same manner the strength of a bar 
of any other substance may be determined, either from direct experi- 
ments on its fiexure, or from the sounds that it produces. If 

♦n /* / 

/ = — , 'ri = -8225 «, and - = v' (8225 «) = 907 ^ n ; whence, 
n a a 

if we know the force required to crush a bar or column, we may 

calculate what must be the proportion of its length to its depth, in 

order that it may begin to bend rather than be crushed. 

When a longitudinal force is applied to the extremities of a straight 

prismatic beam, at the distance D from the axis, the deflection of the 

middle of the beam will be 

"•{-•-(V^-^)-^} ^"""-^ 

If a column, subjected to a longitudinal force, be cut out of a 
plank or slab of equable depth, in order that the extension and com- 
pression of the suifaces may be initially every where equal, its outline 
must be a circular arc. 



CHAP. VIII.] MODULUS OP ELASTICITY. 383 

If a column be cut out of a plank of equable breadth, and the 
outline limiting its depth be composed of two triangles, joined at 
their bases, the tension of the surfaces produced by a longitudinal 
force will be ererj where equal, when the radius of curvature at the 
middle becomes eaual to half the length of the column ; and in this 
case the curve will be a cycloid. 

When the curvature at the middle differs from that of the cycloid, 
the figure of the column becomes of more difficult investigation. It 
may, however, be delineated mechanically, making both the depth of 
the column and its radius of curvature proportional always to ^/a. 
If the breadth of the column vary in the same proportion as the 
depth, they must both be every where as the culw root of a, the 
ordinate. ( Young 9 Nat, Phil, vol. ii.) 

The modulus of elasticity has not yet been ascertained in reference 
to so many subjects as could be wished. Professor Leslie exhibits 
several, however, as below. That of white marble is 2,150,000 feet, 
or a weight of 2,620,000 pounds avoirdupois on the square inch ; 
while that of Portland stone is only 1,570,000 feet, corresponding 
on the square inch to the weight of 1,530,000 pounds. 

White marble and Portland stone are found to have, for every 
square inch of section, a cohesive power of 1811 lbs. and 857 lbs. ; 
wherefore, suspended columns of these stones, of the altitude of 1542 
and 945 feet, or only the ld94th and 1789th part of their respective 
measure of elasticity, would be torn asunder by their own weight. 

Of the principal kinds of timber employed in building and car- 
pentry, the annexed table will exhibit their respective Modulus of 
Elasticity, and the portion of it which limits their cohesion, or which 
lengthwise would tear them asunder. 

Teak 6,040,000 168th 

Oak 4,150,000 144th 

Sycamore 3,860,000 108tli 

Beech 4,180,000 107th 

Ash 4,617,000 109th 

Elm 5,680,000 146th 

Memelfir 8,292,000 205th 

Christiana deal .... 8,118,000 146th 

Larch 5,096,000 121th 

The following, also, exhibits Mr. Sevan s results as to the modulus 
of elasticity. 

Feet. 

Platinum 2,390,000 

Gold (pure) 1,390,000 

Steel 9,300,000 

Bar iron 9,000,000 

Ditto 8,450,000 

Yellow pine 9,150,000 

Ditto 11,840,000 



384 MODULUS OF ELASTICITY. [PART II. 

Feet. 

Finland deal 6,000,000 

Mahogany 7,600.000 

Rose wood 3,600,000 

Oak, dry 6,100,000 

Fir bottom, 25 years old 7,400,000 

Petersburg deal 6,000,000 

Lance wood 5,100,000 

Willow 6,200,000 

Oak 4,360,000 

Satin wood 2,290,000 

Lincolnshire bog oak 1,71 0,000 

Lignum Vitae 1,860,000 

Teak wood 4,780,000 

Yew 2,220,000 

Whalebone 1,000,000 

Cane 1,400,000 

Glass tube 4,440,000 

Ice 6,000,000 

Limestone. 

„ Dinton, Buck 2,400,000 

„ Kctton 1,600,000 

„ Jettemoe 636,000 

Ryegate 621,000 

Yorkshire paving 1 ,320,000 

Cork . 3,300 

Slate, Leicestershire 7,800,000 

The following is the weight of the modidus of elasticity of various 
substances employed very generally in construction, according to Mr. 
Tredgold. 

Pounds. 

Ash 1,640,000 

Beech 1,346,000 

Brass, cast 8,930,000 

Elm 1,340,000 

Fir, red or yellow 2,016,000 

Fir, white 1,830,000 

Iron, cast 18,400,000 

Iron, malleable 24,920,000 

Larch 10,740,000 

Lead, cast 720,000 

Mahogany 1,696,000 

Oak, good English 1,700,000 

Pine, yellow American 1,600,000 

Steel 29,000,000 

Tin, cast 4,608,000 

Steel, cast 13,680,000 



CHAP. Vni.] TaAN8V£B8B 8TBBN0TH OP CAST IBON. 385 



Sbct. IV. On the Strength of Materials svhjected to a tramverse 

strain. 

From the obseirations which we have made in the first section, it 
appears that we are not possessed of all the data which are required 
for determining generally the strength of a cast-iron heam of any 
given form of section ; and consequently, that although rules have 
been given for that purpose, their results are in many cases, (as might 
be expected,) at utter variance with the actual strengths as obtained 
by experiment. Such, then, being the state of our knowledge upon 
the transverse strength of materials, we have abstained from giving 
any rules deduced alone from theory^ being well assured that to have 
an erroneims rule is much worse than to be without ani/ rule at all. 
Nevertheless, the strengths of a few forms having been determined 
by actual experiment, we are enabled to calculate with certainty, by 
means of formula (HI.), page 374, the strength of a beam of any 
dimensions, whose form of section is similar to any of those forms, 
using the word similar here in its strict geometrical sense, as explained 
in definition 12, page 103. 

1. For the rectangular form of section :— From the mean of 265 
experiments of Messrs. Hodgkinson and Fairbairn, it appears that a 
weight of 454*4 lbs. applied at the center of a bar of cast iron 1 inch 
square and with a bearing of 4*5 feet produced fracture ; therefore, 
for a bar of any other dimensions, putting W for the breaking weight 
applied in the center, L the length of the beam in feet, h and d its 
breadth and depth in inches, we have 

2045 6rf' _ . ,^ 
= W, m lbs. 



L 

18'25^(/^ 
L 

•912 b d' 



= W, in cwts. 
=s W, in tons. 



(VIII.) 



2. For Mr. Hodgkinson's form of section (fig. 235) in which the 
*i%a (a) of the lower flange is made six times that of the upper, wc 
We 

4862 ad _ . ,, 
= W, m lbs. 



L 

48-33 acf 
L 

2166a</ 



= W, in cwte.J (IX.) 

= W, in tons. I 



S8S TRAirSVEBSB 8TBSN0TH OF CAST IRON. [PART II 

3. When the form of section does not materially differ from that 
shown in figure 235, the following formula is found hy Mr. Hodg- 
kinson to agree moderately well with the results of experiments. If 
h^ equal the entire hreadth of the bottom flange, b^ the thickness of 
the vertical part, d^ equal the whole depth of the girder, d^ the depth 
without the lower flange, all in inches, and L equal the length in 
feet, we have 

^^^^ {^ ^i' - (*i - ^i) ^1 = W, in lbs. "* 



<L 



40 

{*i ^i' - (^ - ^2) ^2*} = W, in cwts. > ... (X.) 






I 



4. We have also been favoured with the following arbitrary for- 
mula by Mr. Dines, which he has found to be tolerably correct in 
all cases where the length of the girder did not exceed 25 feet, its> 
depth in the center was not greater than 20 inches, nor the breadth 
of its bottom flange less thim one-third, nor more than half th&- 
depth, and the thickness of the metal not less than -ji^th of th^s- 
depth. Then the letters expressing the same quantities as Hpfnw* 
we have 

170^ "V 

-j- {«, d," - (i. - b,) dn = W, in lbs. 

80 

5l7 {*i •'i* - (*, - *,) ''/} = W, in cwU. ... (XI.) 

~ {4, rf.» - (J. - h) «/«*} = W, in toBs. 



LONOrrUDINAL FORM OF BEAM OF UNIFORM STRENGTH. 

1. I/the depth (^tke beam is uniform: — 

When the whole load is collected in one pohit, the sides of th^' 
beam should be straight lines, the breadth at the ends being half thi^^ 
where the load is applied, as in fle. 286. 

When the load is uniformly dntributed, the sides shookl be pos^^ 
tions of a circle, the radius of which should equal the square of tli^v 
length of the beam, divided by twice its breadth, as in fig. 237. 

2. When the breadth of the beam i» uniform: — 

When the load is collected in one point, the extended side of ttr^ 
beam should be straight, the depth at the point where the load ^ 
applied twice that at we ends, and the linea connecting them straight, 
as in fig. 238. 



CHAP. VIII.] TBAN8V£R8B 8TBEN0TH OP MATERIALS. 



387 



When the load is uniformly distributed, the extended side should 
be straight, and the compressed side a portion of* circle whose radius 
equals the square of half the length of the beam divided by its depth, 
as in fig. 239. 

3. When the transverse section rf tie beam is a similar figure 
tknmghout its whole length: — 

When the load is collected ib one point, the depth at this point 
should be to the depth at hm extremities as 3 : 2 ; the sides of the 
beam being all straignt lines, as in fig. 240. 

When the load it vniformly distributed, the depth in the center 
should be to tba depth at the end as 3 : 1 ; the sides of the beam 
being all stnHJgbt lines, as in fig. 241 . 



mANSYKBSE STBENOTH OP OTHER MATERIALS THAN CAST IRON. 

The only form of beam which is employed of any other material 
than cast-iron is the rectangular form, the strength of which may be 
immediately obtained from the following formula, in which h is the 
breadth, d the depth, both in inches, L the length in feet, and W the 
breaking weight applied in the center ; then 

?i^ = W,mlb8. 

li 



L 

o^bd' 



= W, in cwts. 
= W, in tons. 



(XII.) 



The following table exhibits the values of the constant coefiicients 
^^ a,, and a.^ according to Professor Barlow, for several different 
i^nds of timber. 



Name of Material. 



Value 
of a|. 



Value 
of a^ 



Value 
of a^ 



Authority. 



brought iron 
Brass, cast ... 

I«ead, cast 

•nn,caat 

Zinc, cast 

Teak 

Poon 

Ash 

Canadian oak. 
English oak... 
I^tch pine..... 



2290 
890 
196 
872 
746 
821 
740 
676 
589 
567 
544 



20-46 
7-96 
1-76 
3-32 
6-66 
7-83 
6-61 
603 
6-26 
4-98 
4-86 



•022 
•397 
•087 
•166 
•338 
•366 
•380 
•301 
•263 
•249 
•243 



Tredgold. 



Barlow. 



c c 2 



388 



ELASTICITY OF BODIES. 



[PABT II. 



Name of Material 



Beeclj 

Dantzic oak 

Adriatic oak 

Red pine 

Mar Forest fir.... 
New Eugland fir. 

Riga fir 

Elm 

Larcb 



Value 
of a,. 



519 
486 
461 
447 
408 
367 
359 
338 
330 



Value 


Valae 




of a^ 


of aj. 


Authority. 


4-63 


•232 


Barlow. 


433 


•216 


jy 


412 


•206 


>1 


400 


•200 


>^ 


3-64 


•182 


99 


3-28 


•164 


99 


3-21 


•160 


99 


302 


•151 


99 


2-94 


•147 


99 



THE STRENGTH OF BEAMS ACCORDIXO TO THE MANNER IN WHICH 
THE LOAD IS DISTRIBUTED. 

In the same beam, the weight which will be required to break it 
depends very much upon the point at which it is applied ; and upon, 
the ^-ay in which it is distributed ; as also upon the manner in whid^ 
the beam is supported. 

1. When the beam is supported at both ends : — 

If the weight which must be applied at the center to ^ 
produce fracture, equals ) • 

Then that which must be applied at any other point, \ 
(where /j and l^ equal the distances of that point from r L' — 
each support, and L the distance between the sup- I ^ I I 
ports,) ii^ill be equal to ; 

And that which would be required if distributed ) o w 
uniformly along the beam, would be equal to J 

2. When the beam is supported only at one end : — 

If the weight is applied at the other, it must be ) i t^ . 
equal to J * ' 

And if distributed uniformly along the beam, it must ) i »r 
be equal to J * 



Sbct. V. Elasticity of Bodies subfected to a transverse strain, 

The deflection of rectangular beams when supported at each exL - 
mity and loaded in the middle, is found by Uie following fonnu -^ 
in which i equals the deflection in inches, Wj the weight in Iba. pr" "^^ 
ducing it, and m the weight of the modulus of elasticity in lbs. ^^^ 
the given material, which is given at pages 38? and 384 ; the oUt^^^ 
letters represent the same quantities as before, tlien we have 

'=i^^ («"■) 



CHAP. VIII.] ELASTICITY OP BODIES. 389 

If tbe weight be uniformly distributed, instead of being collected 
in tbe center, it will only produce |th8 of tbe deflection given by tbe 
above formula. 

For cast-iron girders whose dimensions are limited, as described 
in § 4, page 386, Mr. Dines finds that the following formulee may 
be made use of to determine the deflection of the beam when 
loaded in the center with |ths of the breaking weight ; in which d is 
the depth of the beam in the center in inches, and L the length in 
feet. 

When the top and bottom flanges are equal, and the girder 
parallel, or of equal depth throughout — 

* = i^ .(^^^-^ 

When the flanges are not equal, and the girder is not parallel — 

* = ^. (^^-J 

When the beam has no top flange, and the depth varies, then 

^ = 8^^ (^VI-) 

Thefte formulee are the result of upwards of 2000 experiments 
upon beams of a moderately large size. 

In the first section we stated that the elasticity of a cast-iron beam 
becomes injured with a strain very much smaller than the breaking 
weight, so that, when the strain is removed, the beam does not 
recover its former shape, but remains permanently bent. Mr. Hodg- 
kinson was, we believe, the first writer who noticed that such was 
the case, which he has proved by his experiments, and which result 
has been fully confirmed by those of Mr. Cubitt. 

The greater part of Mr. Hodgkinson's experiments were made 
apon beams having only a bearing of 4 feet 6 inches ; and he found 
that the amount of the permanent set varied as the square of the 
weight applied. This rule, however, although correct for beams 
about 5 feet in length, does not apply when the length becomes 
mnch greater, for Mr. Cubitt found by his experiments, that when 
the length became equal to about 20 feet, the set was only as the 
weight, and that with larger beams the set was still less. As these 
exponents are important from the size of the girders, we have 
tabulated a few of them, as on the following page. 



390 



MR. GUBITT8 SXPBBIMBNTS. 



[part II. 



1 n 


n 


1 i ij 
1 ip 


1 


BBM^ftKa. 




L 


2 

i 

8 
10 

14 

18 


"000 
172 
-206 
'30fi 
■4B0 

-iia4 

7S2 

sa4 


■100 ^ 0070 

■100 Zim '^^ 
■006 ,S.m ' ^^^ 

102 ^]l mm 


The form and dimeii«Joni of 
theae girdezi are ahon*!! in %« 
242. The result here gtTen are 
tbe mean of two gif4et%, one of 
which weighed 41 cf»t, qn. 20 
Ibs.^ Attd the other 41 cwt. 2 qrt. 
6 Ibft. The (otU length of the: 
girders ww 30 fl. 4 in., the kngth 
of bearing 28 ft. 




^ 


3 
4 

10 
12 
14 
IG 
IB 


Ofll 
121 
■221 
305 
'300 

-ao3 

005 
'70» 
300 


oil i^lJS 

'^'^^ j*0800 


'0050 1 Vm% from tUe wmje jmntsni Uj 
0050 No.t The rwulta are the moml 
-0045 uf two girdent, one of which 
<W70 weighed 42 cwu qn. 14 IbfcJ 
0105 the other 41 cwt. 3 qrt. 16 Ibt^ 
0105 The length of beftring wma 37 £tM 

11165 ?«**^ 1^ 


3. 


2 
4 
6 
8 
10 
12 


•117 
213 

'406 


096 
133 
•120 
-124 

•lie 


-020 
025 
•032 
04$ 
•057 
-068 


^2 The result* arc the oieAn of two 
jjri girders, One of which ireighed 26 
^\l Wt. Iqr. 26 lb*,, and the olher 
;?{* ,25 cni. 2 qm 22 lb«. The tengtb 
"* * |of the girders wm 24 ft, 6 In., the 
ilen^ o( beanng 33 ft 10 in. 


^^ 


4. 


% 
4 
6 
B 
10 
11 


125 

'374 
H03 
012 
071 


'140 

■log 

110 
110 

m9 


0150 
0245 
0350 

mm 

'0620 

mm 


1 Cam from the wiQe pfttUifn ■« 
-0095 'Nil. 3. The faults Are the ni«ftn 
■0105 of two girdeiTft, one of vhich 
■0130 weighed 2&cm, 2 qr«, Iht, uid 
0140 the oiher 24 cwt. 3 qn, I7 Ibsk. 
0070 The lengtb of the be»riBg vm 

|23 ft. 


ft. 


4 



a 

10 
13 


'24 
'38 
'50 
'64 
74 


12 
-14 

14 
•10 


•035 
050 
065 
•HBO 


1 The fonn and d]aie»i»k»ni of tb^^^ 
-01 ft prd*r ajneihown in hg. 244, Th* ^^ 
rtiR 1 weight of the giMer wm 10 cwt*^ ^^ 
;?{* (0 qr». \B lbs., its total length ^r^ 
^*^ Ifeet, and the distftnce betvup. ' ■ 

[the bcftrLngm 20 ft. 7 iru 


6. 


1 

2 
3 
4 
4i 


085 
4:4 
380 1 
402 
475 1 


-0«9 
100 
123 
073 


018 
-035 
•000 
•103 

013 ' 

m& 
■000 ; 


■017 
043 
1123 


E«ctiuiguliir bar of out ln»=3 
51 inches in depth* 1^04 inditf 
in breadtht and 5 ft, 6 in, lengC- 
of bearing. Broke at & tons. 


f 


7. 


1 

3 
4 

44 


083 
16! 
360 
'300 
'440 


•078 
OtfO 
120 
•060 


015 
1Ki2 
•030 


RecL^ngulELr bur of emm iraiiw 
Bi incliei to depth, I 06 incMr 
in br^dth, and 5 feel 6 ipcha 
lengUi of b«mng. Broke v%^ 
41 tonA^ 



CHAP. VIII.] 



MR. CUBITTS EXPERIMENTS. 



391 



i 

d 


1^ 


^1 

11 


Q 


h 

I*' 


1 

a 


mmrABKii 


' 1 
2 
3 

4 
ft 
6 




11*7 
100 
146 
105 
•250 
350 




063 
045 
060 
066 
100 


-005 
-009 
■020 

■lao 


■004 
■Oil 
■160 


RecUngulAr tmr of wucmght 
Iron, 4 '97 mdies in depth, 1 inch 
in breath, and 5 ft 6 in. length 
of be&ring. 


1^ 
1 1 


1 
S 
3 
4 
ft 
6 




060 
110 
166 
210 
300 
630 




*^* No tat, 
0^ i 000 
OfrS 1 012 
090 -5541 

230 ! -260 


■006 
il3a 

■2O0 


Kecungulfir b^r of wrotight 
iron, 5 '03 incbeii in depth* 1 IqcIi 
in breadth. And 6 ft, 6 in* lin^ 
of bearings 


11K 


S 
3 

4 
G 




100 
146 
•206 
402 

but 


-046 
060 

197 
-130 


Trifling 

■01 a 

■154 

'280 


■013 
130 
■126 


Rectangul^ bar of wrought 
iron, 6 inc&Bi m depth, '97 incheii 
In breadth, and 6 ft* 6 in. length 

of bearing. 


11. 


3 
3 

4 . 
ft 




loe 

160 
318 
310 
380 


■062 

■oeo 

-093 
*070 


Trkfling 
■0O8 

■044 
•106 


■012 
<>24 
il6l 


Hecuingular hsr of wrought, 
iron, 4'97 inches In depth, 102 
inchefl iti bresfitbi aod 6 It. 6 In, 
length of bearingp 


IS, 


1 
3 
3 

' 4 




•060 
106^ 
160 
-200 
-238 

26a 


■066 
i)46 
■066 
-032 
■030 


Set. 
004 
■008 
■016 
■020 
■028 


0O4 
■007 
-006 

008 


Rectangular bar of wrought 
iron, 4 97 mckim in depth, 1^02 
inches in breadth, and 6 fu 6 in. 
length of bearing. 


13. 


1 
3 
3 
4 

41 


060 
103 
148 
•200 
-230 


-063 
•046 
■062 
•020 


Set 
■004 




BectJingular bar of wrought 
iron, 4'»7 *nche« in depth, 102 
inchei in breadth, and 6 fU 6 in, 
length of bearing. 


14. 


I 

li 

3 

n 

3 

Si 
4 


•162 
■302 
'348 
■300 
■374 

ma 


■060 
046 
062 
074 


■002 
■003 
008 
■010 
■066 
•306 


•001 
005 
•002 

046 
■161 


Rectangular bar of wrought 
iron, 4 Inches in deplh^ 1^01 
inches in brewJth^ and 6 f^. 6 in, 
length of bearing. 


IS. 


I 

i' 

4 
41 


■100 
150 

loa 

-296 
'3^ 
■460 
G&8 


4)40 
046 

•oeo 

■OGO 
■006 
'208 


^103 
•010 
■016 
■023 
046 
102 
■266 


mi 

006 

■007 

■023 
•066 
163 


Rectangular bar of wrought 
iron, 4 incbea in depth, I -01 
Ini^ei in breAdth, and 6 ft 6 in. 
length of bearing. 



392 



STRBNOTH OP MATERIALS. 



[part II^ 



Sbct. VI. Strength of MaieriaU to resist Torsion. 

The power of a bar or rod of any material to resist torsion, maj^ 
be measured by the angle through which the end of a lever attacbec^ 
to the same will be moved by a given weight. If D equal the dis^ 
tance from the fixed end of the bar or prism to the point of applica^i 
tion of the lever used to twist it, / the length of the lever in inches 
w the weight in lbs. applied to the end of the lever, r the radius c^ 
the pnsm if cylindrical, and 5, d^ its breadth and depth if rectangulau4 
all in inches, § the angle of torsion at the point of application, F a cona 
stant for each material, representing the specific resistance to flexur . 
by torsion, and W a constant weight expressing the resistance to to ^ 
sion, with regard to a unit of surface at the time of fracture ; th^ 
the following table will exhibit the relations subsisting between thc^ 
quantities. 



Form of section of 
the prism. 



Ketistanoe to angular flex- 
ure by a force of torsion. 



Resistance to fracture I 
a force of torsion. 



Cylindrical . 
Square .... 
Rectan":ular 



F = »/ 



2L 



W 



F=«/i(f-±^)i^ 



wl 



2_ 

e 






The mean value of W for cast iron, as obtained from the experi- 
ments of Messrs. George Rcnnic, Bramah, and Dunlop, is 32,503 lbs., 
and from Mr. Bevan's experiments, the mean value of F for cast iron 
is 5,709,600 lbs., and for wrought iron and steel J 0,674,540 Iba^ 
and, these values being substituted in the above table, it becomes 



PonnorMclion 
of the prism. 



Cylindrical 

Square 

Rectangular 



Retlftance to angular flexure by a force of tornoo. 



. Cast iroD. 



u»/ -- 8968620— . 



tr/-96I600 



d*0 



v/-» 1903200 



b*dU 



(6« + rf«)L 



Wrought iron or steeL 



ir/= 16767770—. 

d*i 
^/- 1779090—-. 



Rcristance to ftactwc' 
aforceoftontoa. 
Cattfaon. 



w/- 51066 r«. 
w/»76Clif*. 



IV/-108S4- 






APPENDIX. 







TABLE OL 




[Ho, Um h. 175802 


t 


s 


4 


5 


« 


7 


S 


1 « 


N. 


ISl 


ISQl 
8876 

llHO 

B*2U 


1734 
8038 


2188 
6468 


2598 
8894 


3029 
7321 


3461 
7748 


3S01 
8174 


100 
1 
2 
3 
4 
6 
8 
7 
8 
9 

no 
1 

2 
8 
4 

8 

7 
8 
9 

120 
1 
2 

3 
4 

5 
8 

7i 

8 

9 

ISO 

1 

2 
3 
4 

5 
6 
7 
3 


140 
1 
2 
3 
4 
5 
6 
7 
8 


m 


0300 
4521 
8700 


0724 
4940 
9116 


1147 
5360 
0532 


1670 
5779 
0047 


1993 
6197 


2415 
8616 


\m 


0361 
4488 
8571 


0775 
4896 
S078 


m 


242S 
6533 


2841 
6042 ' 


3252 
7350 


3664 
7757 


4075 
8164 


1^5 
*S7 


060O 
462S 

8620 

3 


1004 
5029 

0017 

4 


1408 

5430 

9414 

5 


18)2 
5830 

9811 
6 


2218 
1230 


2619 
6629 


3021 
7028 


£23 

8 1 


0207 

7 
4148 

8053 


0602 

8 
4540 

8442 


0098 


4038 
8830 


M 
M 


2578 
6495 
03S0 
4^230 
3048 


2989 
6885 


3362 
7275 


8755 
7664 


{46 


0768 
4613 
8428 


1153 
4006 
8805 


1538 
5378 
9185 


1024 
5760 
0563 


2309 
6142 
0042 


2694 
6524 ' 


M»ti 


0320 
4088 
7816 


152 

106 


1829 
5580 
9298 


2206 
5953 
9658 


2582 
8326 


2058 
8609 


3333 
7071 


3709 
7448 


m^ 


0038 
3718 
7363 

5 


0407 
4085 
7731 

6 


0776 
4451 
8004 

7 1 


1145 

4818 
8457 

8 


1514 
5182 
8819 




m 
rre 

8 


29B5 
6640 

3 


3352 
7004 

4 


m 


0206 
3881 

7426 


0826 
4210 
7781 


0087 
4576 
8138 


1347 
4034 

8400 


1707 
5201 
8848 


2067 
5647 

0198 


2428 
6004 
0552 ' 


Ul 
22 


0983 
4471 

7051 


1315 
4820 
8208 


1667 
5160 
8644 


2018 
5518 
8990 


2370 
6866 
9335 


2721 
6215 
9681 


3071 
6562 


HI4 


0028 
3462 

6871 




1403 
4828 
8227 


1747 
5180 
8585 


2001 
5510 
8903 


2434 

5851 
0241 


2777 
6191 
9570 


B119 
6531 

9018 


m» 


0253 
3609 


6040 


:63 

8 

in 


1509 

8 
4644 

8265 


1934 

4 

5278 
8596 


2270 

5 

5811 
8926 


2605 

6 

5943 
9256 


2940 

7 
6276 
9586 


3275 

B 

6608 
9915 


m 


0245 
3525 
6781 


mi 

m 


1560 
4830 
8076 
12&8 
4406 
7671 


1888 
5156 
8300 


2216 
5481 
8722 


2544 

5S06 
9045 


2871 
8131 
9368 


8198 
6456 
9600 


£»3 


0012 
3219 
6403 
9564 


,77 
77 


1819 
4814 
7087 


1030 
5133 
8303 


2260 
5451 

8618 


2580 
5760 
8034 


2000 
8086 
9249 


m 
m 

48 


0822 
3051 

3 

7058 


1136 
4263 

4 

736T 


1450 
4574 

6 
T676 


1763 

4835 

6 
7085 


2076 
5196 

7 

8204 


2380 
5507 

8 
8603 


2702 
6818 

9 

8911 


m 

»43 


0142 
3205 
6246 
9266 


0440 
3510 
8549 

9567 


0756 
3815 
8852 
9868 


1063 
4120 
7154 


1370 
4424 

7457 


1676 
4728 
7750 


1082 
6032 
8061 


»66 


0168 
3161 
6134 
9086 


0489 
3480 
6430 
9S80 


0769 
3753 
6728 
0874 


1068 
4055 
7022 
0988 


m 

'OS 


2266 
5244 
8203 


2564 
6541 
8497 


2863 
58S8 
8792 


148 
eft 


IHl 
4080 


H34 
4351 


1726 
i 4641 


2019 
14932 


2311 


2803 
6512 


2696 ; 
5803 



*B 



K(>. l&OO L. 170091] 




TABLB III. 




[N«. 1999 L. 30031 


1 


m 

1 

s 

3 

* 

9 

160 

1 

, 2 

a 

4 

1 5 





1 


2 


3 


4 


S 


6 


r 1 


a 


e 


s 


176091 


6381 


6670 
0652 


6969 
9839 


724S 


7536 


7825 


8113 


8401 


8689 


15 


8977 i*ao* 


0126 
2985 
5825 
8647 ' 


0413 

3270 
0103 
8928 


0609 
35^5 
6391 1 
0209 


0986 
38^9 
6674 
9490 


1272 

4123 
6956 
9771 


1568 
4407 
7230 


1; 
2 
S 
4 
5 
« 
7 

SOD 
1 
2 

1 

S 

! 

i 
1 

» 

7* 
1 
1 

a' 


1S1S44 
4 SOI 
7521 


2129 
4975 
7803 


2415 
6259 
80S4 


2700 1 

5542 

8366 


msi 

2346 

5623 
£382 


190332 
3125 

8667 


0612 
3403 
6170 


0892 
6453 


1171 

3959 
6720 
H81 


1461 

4237 
7005 
9755 


1730 
4514 
7281 


2010 

4792 
7556 


2289 : 2667 
6009 1 5346 
7832 8107 


8932 


9200 


0029 
2701 

6 
6475 

8173 


0303 
3033 

6 

5746 
3141 


0577 ; 0S5O 


11^4 
3346 f 

.^' 

924TI 


a*l397 



4120 
6326 


16T0 

1 

4391 
7096 


1»43 

2 
466S 
7366 


2216 

% 
4934 
7634 


2488 

4 

6204 
7004 


3305 

7 
6016 
8710 


^77 

8 
6236 
3979 


9515] Ui&a 


0061 


1*319 
2980 
5638 
8273 


0586 
3252 
5902 
8530 


0353 
3518 
6160 

3793 


1121 
3783 
6430 
9U0O 


1368 
4049 
6694 

0323 


1054 
4314 

6957 
9535 


IU21 ! 
45711 '' 
7221 
9845 


212lB8i 2464 

4844' SI 09 
7484 7747 


2720 
5373 
8010 


! 6 
7 
S 


,220108 0370 0631 

1 2716 2970 \ 3236 

£J09 5503 1 5826 


0892 
3490 

0U84 
8667 


1153 
3755 
6342 
3913 


1414 
4U15 
6000 
9170 


1075 
4274 
6858 
9420 


193«j 
4533 

7115 
9682 


2190 
4792 
7372 
0938 


245ff 
5051 
76S0 


9 

m 

1 

3 


7887 1 8144 1 8100 


am . 

1 ' 
2742 1 
5276 

7793 , 


1 

230440 0704 
2996; 3250 
£528 5781 


2 

OOtiO 
3604 
6U33 
8548 


3 
1215 

3767 
6285 
8799 


4 

1470 
4011 
6537 
9049 


5 
1724 
4264 
6789 

9299 




1979 

4517 

7U41 

1 9550 


7 
2234 

4770 
7292 

0800 


i 
243S 
5023 
7544 


8046' 8297 


0050 
2541 
5lM9 

7482 
9032 


om 


4 

e 

7 


240549' 0799 
3038 3280 
6513 6759 
7^73 8219 


1U48 
3534 
60U6 
8464 


1297 
3782 
6252 

8709 


1540 
4030 
6499 
8954 


1795 
4277 
6745 
0198 


2044 

4525 
6991 
9443 


2293 
4772 
7237 
9087 


27*6 1 
5M J 




8 
ISO 

1 


25U420 
2853 

< 
5273 
7679 


U604 
3096 

I 
&614 

7918 


0^08 
S338 

2 
5756 
8158 


1161 
3580 

3 

6090 
3398 


1395 
3322 

4 
6237 

8637 


1638 
4064 

6 
6477 

8877 


1081 
4306 

6 
0713 
9116 


2l:£5 i 2:163 
4548 1 4790 

7 ! « 

6963 7193 
9355 9594 


20l«) 
5031 

9 
7431? 


a 

m 
I 

. t 




i 


260071 
2461 
4818 
7172 
0613 


0310 0548 
2688 1 2925 
6U64 52^0 
7400 7641 
9740 9080 


0787 

3162 
5525 

7876 


1025 
3399 
5761 

3110 


1263 
3630 

5990 
3344 


1601 
3873 
6232 
8678 


1739 1970 
4109 4340 
6467 6702 
3812 1 9040 


1214 




0213 
2538 
4350 
7151 

3 
9439 


iHl46 
2770 
50B1 
7330 

; 4 
9067 


Otfr9 
3001 
5311 
7609 

5 

9895 


0912 
3233 
5542 

7838 

6 


1144 
3464 
5772 

8067 

7 


1377 

3606 
0002 i 
S296 

8 


im 




V 
8 
9 


271842 

4158 
6163 




2074 
4389 
0692 

1 


2ay(J 
46:i0 
0921 

2 


39iT 






190 


8754 


8932 ; 9211 


0123 
2306 

4056 
6905 

9ua 


0351 

2022 
4832 
71 SO 
9366 


0578 1 
2849 
6107 
T354 

9539 


"«Wi 


m 




1 
t 
3 


2810331 1201 ' 1488 
3301 3527 ^753 
5657| 5782 6007 
7S02I 8023 8249 


1716 

3979 
6232 
8473 


1942 
4205 
6456 

8696 


2169 
4431 

6631 
3020 


m 
m 

m 


1 
i 

i 

i 

1 


i 
1 


5 



7 

S 
9 


290OJJ5 
2266, 
4400 
6666 

8863, 


0267 
2478 
4687 

68^4 i 
0071 


0480 

mn 

4907 
7104 
0289 


0702 
2920 
5127 1 
7323 
9507 


0925 
S141 
6347 
7542 
9725 


1147 
3363 
5567 
7761 
0043 


1309 
3584 
5787 
7979 


1591 
3304 
6007 
3198 


1313 
4026 
0226 

8416 




i 
1 

t 


0161 


0373 


0696 


1 oau 


1 



V%i0io 1^0010301 




TABLB IIL 




[N*. 2499 J,. 897766 


1 1 

2 
3 
4 

6 

1 T 
& 

9 





1 


S 


8 


4 


6 


6 


7 


8 


9 


K. 


301030 
S19@ 

7490 

M30 

311754 

S3&7 

5^0 


1247 
8412 
8566 
7710 
0843 

wm 

4078 
6180 
8S72 


1464 
3628 

8781 
7924 


1681 
3844 
5996 

8137 


1398 
4069 
6211 
8361 


2114' 
4275 
6425 
8564 


2331 
4491 
6639 
8778 


2647 
4706 
6854 
8601 


2764 
4921 
7068 
9204 


2980 
5136 
7282 
9417 


2oa: 

li 

2 
3 
4 

5 
6 
7 

8 

6 


0066 
2177 
4239 
6390 
8481 


0263 
2380 , 
4490 
6699 
8689 


0481 
2600 
4710 
6809 
8808 


0693 
2812 
4020 
7018 
9106 


0906 
3023 
5130 
7227 
6314 


1118 
3234 
5340 
7436 

9522 


1330 
3446 
5561 

7646 
6730 


1542 
3656 
5760 
7854 
9938 


320146 


0354 


05^2 j 


0769 


0977 


1184 


1391 


1593 


1805 


2012 


1 
2 
3 
1 

5 

! d, 
r 



2210 

4282 
6336 


1 
2426 

4488 
6541 


2 
2633 

4694 
6746 
8787 


3 
2839 
4869 
6950 

8991 


4 

3046 

5105 
7155 
9164 


5 
3252 
5310 
7359 
9308 


6 
3458 
5516 
7563 
9601 


7 

3665 
5721 

7767 
0806 


8 
3871 
5926 
7972 


9 

4077 
8181 
8176 


210 
1 
2 


l^3S0 nsh^ 


0008 
2034 
4051 
6059 
8058 


0211 
2236 
4263 
6260 
8267 


3 


330414 oeir ; 

2438 S64(» 
4154 16^ 

6460 6660 
84^6 86M 


0819 
2842 

4856 
6860 

S855 


1022 
3044 

6057 

roflo 

9064 


1226 

3246 
6257 
7260 
9253 


1427 
3447 

6458 
7459 
9451 


1630 
3649 
6658 
7659 
6650 


1832 
8860 
5869 

7866 
9849 


41 
6 
6, 
T 


U047 
2028 


0246 


fl 


S40444 064^ 


0841 


1030 


1237 


1436 


1632 


1830 


2226 9 


1 
s 

1 

! 




2423 

43&2 

e353 
S30j^ 


1 
2820 
4689 
6549 
8600 


2 

2817 
4786 
6744 
8094 


3 
3014 
4081 
6039 
8SS9 


1 

3212 
5178 
7135 
9083 


5 
3409 
6374 
7330 
9278 


6 
3606 
5570 
7525 
0472 


7 
3802 
5766 
7720 
9666 


8 
3969 
6662 
7615 
9860 


9 
4196 2^0 
6167 1 1 
8110 1 a 

0064 3 


350-^48 
218S 
4108 

eo2« 

7935 
»835 


0442 
S375 
4301 
8217 
81 35 


0636 

2668 
4493 
6403 
8316 


0829 
2761 

4686 
6699 
8606 


1023 
2654 
4876 
6790 
8696 


1216 
3147 
6068 
6981 
8886 


1410 
3339 
6260 
7172 
6076 


16i>3 
3632 
5462 
7363 
9266 


1708 

3724 
5643 
7554 
9456 


1689 
3910 
5834 
7744 
9646 


4; 

6 
6 

7 
8 
9 




oose 


021^ 


040^ 


0593 


0783 


0972 


1161 


1350 


1539 


L 

1 
3 

3 


1 
301728 1»17 
M12: 8800 
W88' fi675 
736« 7543 
9218 9401 


210# 
8938 

6862 
77^ 
9687 


3 
2294 
4176 

8049 
7015 
©772 


4 
2482 
4363 

6236 
8101 
9658 


6 
2871 
4551 

6423 
8287 
0143 
1091 
3831 
5664 
7488 
9808 


6 
2859 
4739 
6610 
8473 
0328 
2176 
4016 
5846 
7670 
9487 


7 

3048 
4626 
6796 
8659 
0513 
2360 
419B 
60-26 
7862 
9668 


8 
3238 

5113 
6083 
8845 
0698 
2544 
438? 
6212 

mu 

9849 


6301 
7169 
6030 
08S3 


330 
I 

3 
4 


• 


37l0fiS 1263 
2912' 3096 
4748 4932 
8677, 6769 
8S98' 8630 


1437 
3280 
5115 
6942 

8761 


1622 
3484 

5298 
7124 
8943 


JbOe 
3647 
5481 
7308 
9124 


2728 
4566 
6364 

8216 


5 

8 


9 


0030 

9 

1837 
3636 
5428 
7212 
8989 


9 

240 
1 
S 
8 
4 

5 
6 

r 

8 
9 






»1T 
8815 

M06 

r»90 


i 

038^ 
2197 
8995 

8785 
75$8 


0673 
2877 
4174 
5964 
7746 
9620 


3 
0754 

2557 
4363 

6142 
7924 
9698 


4 

0634 
2737 
4533 
6821 
8101 
0876 


5 
1115 
2917 
4712 
6499 
8279 


6 
1296 
3097 
4891 
6677 
8456 


7 
147.6 
3277 
5070 
6356 
8634 


8 
1666 
3456 

6249 
7034 
8811 


\ ' 


9166 9348 


0051 
1817 
3575 
6326 
7071 


0223 
1993 
3761 
5501 
7245 


0406 
2166 
3936 


0582 
2345 
4101 
6850 
7592 


0769 
2521 

4277 
6026 
7766 


y 


39093^ 1112 
i8»7 2878 
4452 4627 
Bim 08T4 


1238 
3048 

4802 
6^8 


1464 
8224 

4977 
! 67i2 


1841 
3400 
6152 
6396 



*Ba 



No. 2600 L. 397940] 



TABLB III. 



[No. 2999 L. 476976 



8461 I 8634 




2 401401 
8 3121 

4 4834 

5 6540 

6 8240 

7 9933 



6710 I 6881 
8410 8579 



0365 
2089 
3807 
5617 
7051 I 7221 
8749 ' 8918 



8808 , 8981 



9154 



0538 0711 
2261 2433 
3978 I 4149 
5688 5858 
7391 I 7561 
9087 ! 9257 



8 411620 
9, 3300 



260 4973 

1 6641 

2 8301 
8 9956 

4 421604 

5 3246 

6 4882 

7 6511 

8 8135 

9 9752 



0102 0271 0440 ' 0609 ! 0777 0946 
1788 j 1956 I 2124 2293 2461 ' 2629 
3467 3035 , 3803 3970 4137 ! 4305 



1 

5140 
6807 i 
8467 ' 



I 



6 



2 ' 3 4 

5307 6474 | 5641 6808 , 5974 

6973 7139 , 7306 I 7472 ' 7638 

8633 8798 8964 , 9129 9295 



0883 
2605 
4320 
6029 
7731 



'N. 



9328 ' 9501 250 

1 



1056 , 1228 
2777 2949 
4492 4663 
6199 ' 6370 
7901 8070 



9426 9595 9764 



1114 1283 
2796 2964 



1451 
3132 



4472 I 4639 ' 4S06 ] 9^ 



8 



6141 ' 6308 , 6474 260 



7804 i 7970 , 8135 
9460 9625 ' 9791 



1 



0121 
1768 
3410 
6045 
6674 
8297 
9914 



0286 0451 0616 



1933 
3574 

5208 



2097 I 2261 
3737 3901 
6371 5534 
6836 I 6999 | 7161 
8459 8621 8783 



0075 j 0236 i 0398 



270 431364' 

1 2969 

2 4569! 

3 6163 
7751 
9333 



4 

6 

6 440909 

7 2480 

8 4045 
9, 5604 



280 7168 
1 8706 



1 
1526 
3130 
4729 
6322 
7909 
9491 



1685 
3290 
4888 
6481 
8067 
9648 



8 
1846 
3450 
6048 
6640 
8226 
9806 



4 

2007 
3610 
5207 
6799 
8384 
9964 



1066 


1224 


1381 


1538 


2637 


2793 


2950 


3106 


4201 


4357 


4613 


4669 


6760 


6915 


6071 


6226 


1 


2 


8 


4 


7818 


7468 


7623 


7778 


8861 


9015 


9170 


9324 



2 450249 

8, 1786 

41 3318 

6 4845 

6 6366 

71 7882 

8 9392 



0865 , 
2400 I 
3930 ! 
5454 
6973 , 
8487 
9995 



9 460898 




290 2898 

li 8893 

2 6883 

8 6868 

41 8847 

6 9822 




0657 


0704 


2026 2171 


8487 


8638 


4944 


6090 


6897 


6542 



. 8000 L. 477121]^ 




TABLE III. [No. 3499 L. 543944 





1 


2 


3 


4 


5 


6 
7989 


7 


8 


1 
9 


N. 
800 


) 477121 


7266 


7411 


7555 


7700 


7844 


8133 


8278 


1 8422 


L 8566 


8711 


8855 


8999 


9143 


9287 1 9431 


9575 


9719 


9863 


1 
2 


I 480007 


0151 


0294 


0438 


0582 


0725 


0869 


1012 


1166 


1299 


\ 1448 


1586 


1729 


1872 


2016 


2159 


2302 


2446 


2688 


2731 


3 


I 2874 


3016 


3159 


3302 


8445 


3587 


3730 


3872 


4016 


! 4157 


4 


> 4300 


4442 


4585 


4727 


4869 


5011 


6163 


6296 


5437 


6679 


5 


5 5721 


5863 


6005 


6147 


6289 


6430 


6572 i 6714 


6866 


6997 


6 


' 7138 


7280 


7421 


7563 


7704 


7845 


7986 


8127 


8269 


8410 


7 


\ 8551 


8692 


8833 


8974 


9114 


9255 


9396 


9637 


9677 


9818 


8 
9 


> 9958 


0099 

1 


0239 
2 


0380 
3 


0620 

4 


0661 
5 


0801 
6 


0941 
7 


1081 
8 


1222 
9 





) 491362 


1502 


1642 


1782 


1922 


2062 


2201 


2341 


2481 


2621 


310 


2760 


2900 


3040 


8179 


3319 


3458 


3697 


3737 


3876 


4015 


1 


I 4155 


4294 


4433 


4572 


4711 


4860 


4989 


5128 


6267 


5406 


2 


I 5544 


5683 


5822 


5960 


6099 


6238 


6376 


6516 


6663 


6791 


8 


^ 6930 


7068 


7206 


7344 


7483 


7621 


7759 


7897 


8036 


8173 


4 


i 8311 
1 9687 


8448 
9824 


8586 
9962 


8724 


8862. 


8999 


9137 


9275 


9412 


9550 


5 
6 

7 


0099 
1470 


0286 
1607 


0874 
1744 


0611 1 0648 


0785 
2164 


0922 
2291 


501059 


1196 


1333 


1880 


2017 


2427 


2564 


2700 


2837 


2973 


3109 


3246 


3382 


3618 


3656 


8 


8791 


3927 


4063 


4199 


4335 


4471 


4607 


4743 


4878 


6014 


9 





1 


2 


3 


4 


5 


6 


7 


8 


9 




5150 


5286 


5421 


5557 


5693 


5828 


5964 


6099 


6284 


6370 


820 


6505 


6640 


6776 


6911 


7046 


7181 


7316 


7461 


7686 


7721 


1 


7856 7991 


8126 


8260 


8395 


8530 


8664 


8799 


8984 


9068 


2 


9203 


9337 


9471 


9606 


9740 


9874 


0009 
1349 


0148 
1482 


0277 
1616 


0411 
1760 


3 

4 


510545 


0679 


0813 


0947 


1081 


1215 


i 1883 


2017 


2151 


2284 


2418 


2561 


2684 


2818 


2961 


3084 


5 


1 8218 


3351 


3484 


3617 


3750 


3888 


4016 


4149 


4282 


4415 


6 


' 4548 


4681 


4813 


4946 


5079 


5211 


5344 


6476 


6609 


5741 


7 


( 5874 


6006 


6139 


6271 


6403 


6535 


6668 


6800 


6932 


7064 


8 


» n96 


7328 


7460 


7592 


7724 


7855 


7987 


8119 


8261 


8382 


9 





1 


2 


3 


4 


5 


6 


7 


8 


9 




> 8514 
I 9828 


8646 
9959 


8777 


8909 


9040 


9171 


9303 


9484 


9666 


9697 


330 

1 
2 


0090 
1400 


0221 
1530 


0353 
1661 


0485 
1792 


0615 
1922 


0745 
2053 


0876 
2183 


1007 
2814 


S 521188: 1269 


\ 2444 


2575 


2705 


2835 


2966 


3096 


3226 


3366 


3486 


3616 


3 


I 8746 


3876 


4006 


4136 


4266 


4396 


4526 


4666 


4785 


4915 


4 


> 5045 


5174 


5304 


5434 


5563 


5693 


5822 


6961 


6081 


6210 


5 


S 6339 


6469 


6598 


6727 


6856 


6985 


7114 


7243 


7372 


7601 


6 


r 7680 


7759 


7888 


8016 


8145 


8274 


8402 


8531 


8660 


8788 


7 


J 8917 


9045 


9174 


9302 


9430 


9569 


9687 


9815 


9943 


0072 
1361 


8 
9 


> 530200 


0328 


0456 


0584 


0712 


0840 


0968 


1096 


1223 





1 


2 


3 


4 


5 


6 


7 


8 


9 




) 1479 


1607 


1734 


1862 


1990 


2117 


2245 


2372 


2500 


2627 


340 


I 2754 


2882 


3009 


3136 


3264 


3391 


8518 


3645 


3772 


3899 


1 


I 4026 


4153 


4280 


4407 


4534 


4661 


4787 


4914 


5041 


5167 


2 


) 5294 


5421 


. 5547 


5674 


5800 


5927 


6063 


6179 


6806 


6432 


3 


1 6558 


6685 


6811 


6937 


7063 


7189 


7316 


7441 


7667 


7693 


4 


S 7819 


7945 


8071 


8197 


8322 


8448 


8574 


8699 


8825 


8951 


5 


Jl^9076 


9202 


9327 


9452 


9578 


9703 


9829 


9954 


0079 
1830 


0204 
1454 


6 

7 


r 540329 


0456 


0580 


0705 


0830 


0955 


1080 


1205 


Bl 1679 


1704 


1829 


1953 


2078 


2203 


2827 


2452 


2576 


2701 


8 


r. 2820 


8950 


3074 


8199 


3323 


3447 


8571 


3696 


9820 


8944 


9 



No. 8500 L. 544068] 



TABLE III. 



[Xo. 8999 L. 601951 



N. 



8 



850,544068 

1 6307 

2 6548 
8 7775 



9003 



4192 I 4816 
5431 I 5555 
6666 6789 
7898 ' 8021 
9126 9249 



4440 
5678 
6913 
8144 
9371 



4 


5 


6 


7 


8 


4564 


4688 


4812 


4986 


5060 


5802 


5925 


6049 


6172 


6296 


7086 


7159 


• 7282 


7405 


7529 


8267 


8389 


8512 


8635 


8758 


9494 


9616 


9739 


9861 


99S4 



9 S. 

5188 350 
6419 1 
7652 
&SS1 



550228 
1450 
2668 
3883 
5094 



860 
1 
2 
8 
4 
5 
6 



0351 0478 
1572 , 1694 
2790 ' 2911 
4004 4126 
5215 5836 



! 1 
6303 6423 
7507 7627 
8709 8829 
9907 "0026" 



2 
6544 
7748 
8948 



0595 
1816 
3038 
4247 
5457 

3 

6664 
7868 
9068 



0717 
1938 
3155 
4868 
5578 

4 
6785 
7988 
9188 



0840 
2060 
3276 
4489 
I 5699 



I 

I 5 
6905 
8108 
9308 



, 0962 
I 2181 
I 3398 
I 4610 
> 5820 

i 6 

7026 
8228 
9428 



1084 
2808 
3519 
4781 
5940 

7 
7146 
8349 
9543 



1206 
2425 
3640 
4852 
6061 



01i« 
1828 
2547 
8762 
4978 
6182 



561101 1221 

2298 2412 

3481 3600 

4666 4784 

8 5848 5966 

9 7026 7144 



870 

! 1 



I 1 
8202 8319 
9874 9491 



0146 
1340 
2531 
3718 
4903 
6084 
7262 

2 

8486 
9608 



0265 
1459 
2650 
8837 
5021 
6202 
7379 

3 

8554 

I 9725 



0385 
1578 
2769 
8955 
5139 
6320 
7497 

4 

8671 
9842 



2 570543 0660 



S80 



1709 1825 

2872 2988 

4031 4147 

5188 5308 

6341 6457 

7492 7607 

8639 8754 




9784 



1 
9898 



580925 
2063 
3199 
4331 
5461 
6587 
7711 
8882 
9950 



0776 
1942 
3104 
4263 
5419 
6572 
7722 
8868 

2 



0893 
2058 
3220 
4379 
5534 
6687 
7886 
8983 



1039 
2177 I 
3312 
4444 , 
5574 
6700 
7823 
8944 



0012 I 
1153 1 
2291 I 



0061 



I 
591065 
2177 
8286 
4393 
5496 
6597 
7696 
8791 
9883 



890 
1 
2 
8 
4 
5 
6 
7 
8 

. 9 e mfsj^^ 



1 

1176 
2288 
3897 
4508 
5606 
6707 
7806 
8900 
9992 



3426 
4557 
5686 
6812 
7985 
9056 



0173 

2 
1287 
2399 
3508 
4614 
5717 
6817 
7914 
9009 



0126 
1267 
2404 
3539 
4670 
5799 
6925 
8047 
9167 



1010 
2174 
8336 
4494 
5650 
6802 
7951 
9097 



0504 
1698 
2887 
4074 
5257 
6437 
7614 

5 

8788 
9959 



0624 , 0743 
1817 I 1986 



' 3006 
i 4192 
I 5376 
6555 
' 7732 



1126 

2291 

. 3452 

; 4610 

! 5765 

6917 

8066 

9212 



0101 



0284 

3 
1399 
2510 
8618 
4724 
5827 
6927 
8024 
9119 



0210 



0241 I 0855 
1381 ! 1495 
2518 2631 
3652 3765 
4783 4896 
5912 i 6024 
7037 ; 7149 
8160 I 8272 
9279 1 9391 



3125 
4811 
5494 
6678 
7849 



6 7 

8905 9023 



8 9 

7267 ' 7337 $60. 
8469 ! 8589 I 
9667 ■ 9787 2 
S 
4 
5 
6 



0863 0982 

2055 ; 2174 

3244 , 3362 

4429 4548 

5612 ; 5780 

6791 6909 

7967 8084 



8 9 

9140 9257 i70i 



0076 
1248 
2407 
3568 
4726 
5880 
7032 
8181 
9326 



, 0193 
I 1359 
i 2523 
] 3684 
4841 
5996 
: 7147 
'■ 8295 
. 9441 



03U9 i 0426 1. 
i 1476 15H2 2 
I 2639 I 2755 3 



3800 
495*7 
6111 
7262 
8410 
9555 

8 



0469 
1608 
2745 
3879 
5009 
6137 
7262 
8384 
9508 



0583 
1722 
2858 
3992 
5122 
6250 
7374 
8496 
9615 



0697 , 
1836 I 
2972 ' 
4105 ! 
5235 
6362 I 
7486 ; 
8608 ; 
9726 ; 



3919 i 4 

6072 5 

6226 «; 

7877 "; 

8625; f 

9^ ^^ 

i 

9 J I 
0811 .8W: 
1960 li 



4218 5| 

5848 4 

6475 5; 

7599 5 

8720 '; 

9888' 9 



0396 0607 



4 
1510 
2621 
8729 
4834 
5937 
7037 
8134 
9228^ 
0819 
1408 



6 
1621 
2732 
3840 
4946 
6047 
7146 
8248 
9337 



0428 
16U 



0619 

6 
1732 
2848 
8950 
5056 
6157 
7256 
8868 
9446 



0687 



0730 

7 
1848 
2954 
4061 
6165 
6267 
7366 
8462 
9566 



0646 
1734 



0842 

8 
1955 
3064 
4171 
5276 
6877 
7476 
8672 

iMAK 
VvOv 



0958 I 9| 

9 

2066 m 

8176 1 

42S2 3: 

6487! 4 
7586= 5 

mi , «< 

9774! 



0766 
l«4k 



0604 
1961 



4 " 



iNoi 4000 L. 



TABLE IIL 



[No. 4499 L. 653116 



N. 



400 602060 2169 

1. 8144 j 8258 

4226- 4834 

5805 5413 

68811 6489 

7455! 7562 

8526 8688 

95941 9701 



8 610660 0767 

9 1728 1829 



410 
1 
2 
8 
4 
5 
6 





2784 
8842 
4897 
5950 
70001 
8048' 
9098 



1 
2890 
8947 
5008 
6055 
7105 
8158 
9198 



2277 
8861 
4442 
5521 
6596 
7669 
8740 
9808 



0878 
1986 

2 
2996 
4058 
5108 
6160 
7210 
8257 
9802 



7 620186 0240 

8 1176 1280 

9 2214 2818 



420 
1 
2 
8 

4 
5 
6 

7 
8 




480 

ii 

4 
5. 
6' 



1 
8249 8358 
4282 4885 
5812! 5415 
6840 6448 
7366; 7468 
8889; 8491 
9410 9512 



0844 
1884 
2421 

2 

8456 
4488 
5518 
6546 
7571 
8598 
9618 



680428, 0580 
1444 1545 
2457! 2559 

1 
8468 8569 
4477; 4578 
5484! 5584 
6488- 6588 
7490 7590 
8489' 8589 
9486 9586 



0681 
1647 
2660 

2 
8670 
4679 
5685 
6688 
7690 
8689 
9686 



7i640481, 0581 
8: 1474! 1573 
9I 2465 2568 



440= 
li 
2! 
8 
4 
5 
6 
76508^ 



I 1 
8453- 3551 
4439 4537 
5422, 5521 
6404' 6502 
7883, 7481 
8360 8458 
98851 9482 



1S78 
ffi46 



0405 
1375 



0680 
1672 
2662 

2 
8650 
4686 
5619 
6600 
7579 
8555 
9530 

0502 
1472 
2440 



2886 
8469 
4550 
5628 
6704 
7777 
8847 
9914 



2494 
8677 
4658 
5786 
6811 
7884 
8954 



I 



0979 
2042 

8 
3102 
4159 
5218 
6265 
7815 
8362 
9406 



0021 
1086 
2148 

4 
8207 
4264 
5319 
6870 
7420 
8466 
9511 



0448 
1488 
2525 



0552 
1592 
2628 



3 4 

3559 3663 
4591 I 4695 
5621 j 5724 



6648 
7673 
8695 
9715 



6761 
7775 
8797 
9817 



0783 0886 
1748 1849 
2761 2862 



8 
8771 
4779 
5785 
6789 
7790 
8789 
9785 



4 

3872 

4880 
5886 
6889 
7890 
8888 
9885 



0779 
1771 
2761 



0879 
1871 
2860 



3 4 
8749 8847 
4784 I 4882 
5717 ' 5816 
6698 6796 
7676 . 7774 
8658 ; 8750 
9627 I 9724 



2608 
8686 
4766 
5844 
6919 
7991 
9061 



2711 
8794 
4874 
5951 
7026 
8098 
9167 



0128 
1192 
2254 

5 
8813 
4870 
5424 
6476 
7525 
8571 
9615 



0284 
1298 
2860 

6 
8419 
4475 
6629 
6681 
7629 
8676 
9719 



0656 
1695 
2782 

5 

8766 
4798 
5827 
6853 
7878 
8900 
9919 



0760 
1799 
2835 

6 
3869 
4901 
6929 
6966 
7980 
9002 



0936 
1951 
2968 

5 
8973 
4981 
5986 
6989 
7990 

8988 

9984 I-QOST 



0021 
1038 
2062 
3064 

6 

4074 
5081 
6087 
7089 
8090 
9088 



0978 I 1077 
1970 2069 
2969 3058 



0599 
1569 
2536 



0696 
1666 
2633 



5 

3946 
4931 
6913 
6894 
7872 
8848 
9821 

0793 
1762 
2730 



6 
4044 
6029 
6011 
6992 
7969 
8945 
9919 

0890 
1859 
2826 



2819 
3902 
4982 
6069 
7133 
8205 
9274 



2928 , 3036 
4010 , 4118 
5089 ! 6197 
6166 6274 
7241 7848 
8312 8419 
9881 9488 



0341 
1406 
2466 

7 
8625 
4681 
6634 
6686 
7734 
8780 
9824 



0447 
1611 
2672 

8 
8630 
46S6 
6740 
6790 
7839 
8884 
9928 



0664 
1617 
2678 

9 
3736 
4792 
6846 
6896 
7943 
8989 



0864 


0968 


1903 


2007 


2939 


3042 


7 


8 


3973 


4076 


6004 


6107 


6032 


6185 


7068 


7161 


8082 


8185 


9104 


9206 



0032 
1072 
2110 
3146 

9 

4179 
6210 
6238 
7263 
8287 
9308 



0123 
1139 
2153 
3166 

7 
4175 
5182 
6187 
7189 
8190 
9188 



0183 
1177 
2168 
8166 



4143 
6127 
6110 
7089 
8067 
9048 



0224 
1241 
2255 
3266 

8 
4276 
6283 
6287 
7290 
8290 
9287 



0326 
1842 
2366 
3367 

9 
4376 
5383 
6388 
7390 
8389 
9387 



0283 
1276 
2267 
3265 



I 



0016 
0987 
1956 
2923 



8 

4242 
5226 
6208 
7187 
8165 
9140 



0382 
1376 
2306 
3354 

9 
4340 
6324 
6306 
7286 
8262 
9237 



N. 



400 

1 
2 
3 
4 
6 
6 

7 
8 
9 



410 
1 
2 
3 
4 
5 

6 
7 
8 
9 



420 
1 
2 
3 
4 
5 
6 
7 
8 
9 



0118 
1084 
2053 
8019 



0210 
1181 
2150 
8116 



430 
1 
2 
3 
4 
5 
6 
7 
8 
9 



440 
1 
2 
3 
4 
5 
6 
7 
8 
9 



No. 4500 L. 053218] 




TABLE III. 




[No. 4999 L 69888 


N. 

1 


1 


2 


3 


4 


5 


6 


7 


8 , 9 jir 


450 653213 


8309 


3405 


3502 


3598 


8695 


3791 


3888 


8984 4080 4C 


1 4177 


4273 


4369 


4465 


4562 


4658 


4754 


4850 


4946 5042 ; 


2 5188 


5235 


5331 


5427 


5528 


5619 


5716 


5810 


5906 6002 ! 


8 6098 


6194 


6290 


6386 


6482 


6577 


6673 


6769 


6864 6960 


4 7066 


7152 


7247 


7843 


7438 


7534 


7629 


7725 


7820 , 7916 . 


5 8011 


8107 


8202 


8298 


8893 


8488 


8684 


8679 


8774 ' 8870 | 


6 8965 


9060 


9155 


9250 


9346 


9441 


9686 


9631 


9726 9821 


7; 9916 


















0011 


0106 


0201 


0296 


0391 


0486 


0581 


0676 0771 ! 


8 660865 


0960 


1055 


1160 


1245 


1339 


1434 


1629 


1628 1718 


9 


1813 


1907 


2002 


2096 


2191 


2286 


2380 


2476 


2669 2663 







1 


2 


8 


4 


6 


6 


7 


8 < 9 


460 


2758 


2852 


2947 


8041 


3135 


3230 


3324 


3418 


8612 3607 m 


1 


3701 


3795 


3889 


8983 


4078 


4172 


4266 


4860 


4464 4648 


2 


4642 


4736 


4830 


4924 


5018 


6112 


5206 


5299 


6893 6487 


3 


5581 


5675 


5769 


5862 


5966 


6060 


6143 


6237 


6331 I 6424 


4 


6518 


6612 


6705 


6799 


6892 


6986 


7079 


7173 


7266 7860 


5 


7458 


7546 


7640 


7733 


7826 


7920 


8018 


8106 


8199 8298 


6 


8886 


8479 


8572 


8665 


8759 


8862 


8945 


9088 


9131 9224 • 


7 


9317 


9410 


9503 


9596 


9689 


9782 


9875 


9967 








0060 : 0153 
0988 ■ 1080 ! 


8 670246 


0839 


0431 


0524 


0617 


0710 


0802 


0895 


9 


1173 


1265 


1358 


1461 


1543 


1686 


1728 


1821 


1913 , 2005 







1 


2 


8 


4 


5 


6 


7 


8 1 9 


470 


2098 


2190 


2283 


2875 


2467 


2560 


2652 


2744 


2836 2929 4: 


1 


3021 


3118 


3205 


8297 


3390 


8482 


3674 


3666 


3758 3850 1 


2 


3942 


4034 


4126 


4218 


4810 


4402 


4494 


4586 


4677 , 4769 ' 


3 


4861 


4958 


5045 


5137 


5228 


5320 


5412 


5503 


5695 1 5687 j 


4 


5778 


5870 


5962 


6053 


6145 


6236 


6328 


6419 


6611 1 6602 1 


5 


6694 


6785 


6876 


6968 


7059 


7161 


7242 


7338 


7424 ! 7516 ' 


6 7607 


7698 


7789 


7881 


7972 


8068 


8154 


8245 


8336 


8427, 


7! 8518 


8609 


8700 


8791 


8882 


8978 


9064 


9155 


9246 


9357 


8! 9428 


9519 


9610 


9700 


9791 


9882 


9973 








9680336 


0426 


0517 


0607 


0698 


0789 


0879 


0068 
0970 


0154 0246 
1060 1151 ; ! 


1 


1 


2 


3 


4 


5 


6 


7 


8 


9 i 


480 1241 


1832 


1422 


1513 


1603 


1693 


1784 


1874 


1964 


2055 48> 


1 2145 


2285 


2826 


2416 


2506 


2696 


2686 


2777 


2867 


2957 


2 3047 


3137 


8227 


3317 


3407 


3497 


3587 


3677 


8767 


3857 


3 3947 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 


4: 4845 


4935 


5025 


5114 


5204 


6294 


5383 


6478 


5563 


5652: 


5 5742 


5881 


5921 


6010 


6100 


6189 


6279 


6368 


6458 


6547 1 


6 6686 


6726 


6815 


6904 


6994 


7083 


7172 


7261 


7351 


7440; 


7i 7529 


7618 


7707 


7796 


7886 


7975 


8064 


8153 


8242 8831 1 


8, 8420 


8509 


8598 


8687 


8776 


8865 


8953 


9042 


9181 


9220' 


9 


9809 


9898 


9486 


9575 


9664 


9753 


9841 


9930 




J 







1 


2 


8 


4 


5 


6 


7 


0019 
8 


0107 
9 




490 690196 


0285 


0873 


0462 


0550 


0639 


0728 


0816 


0905 


0993 


49 


1' 1081 


1170 


1258 


1847 


1485 


1524 


1612 


1700 


1789 


1877 




2 1965 


2053 


2142 


2230 


2318 


2406 


2494 


2588 


2671 


2759 




3 2847 


2935 


3023 


3111 


3199 


3287 


3375 


8468 


3551 


3639 




4 8727 


3815 


3903 


8991 


4078 


4166 


4254 


4842 


4430 


4517 




5 4605 


4693 


4781 


4868 


4956 


5044 


5131 


5219 


5307 


5394 




6 5482 


5569 


5657 


5744 


5832 


5919 


6007 


6094 


6182 


6269 




7 6356 


6444 


6581 


6618 


6706 


6798 


6880 


6968 


7055 


7141 




8 7229 


7817 


7404 


7491 


7578 


7665 


7752 


7889 


7926 


8014 


1 


9 8IOO: 8188 


8275 


8362 


8449 


8585 


8622 


8709 


A7M 


8888 1 



N. 6000L. 


698970] 






TABLE 111. 




[No. 5499 L. 740284 


-•1 

500 
1 





1 


2 


3 


4 


5 


6 


7 


8 


9 


N. 


698970 
9838 


9057 
9924 


9144 


9231 


9817 


9404 


9491 


9578 


9664 


9751 


500 

1 
2 


0011 
0877 


0098 
0968 


0184 
1050 


0271 
1136 


0358 
1222 


0444 
1309 


0531 
1395 


0617 
1482 


2 700704 


0790 


3 1568 


1654 


1741 


1827 


1913 


1999 


2086 


2172 


2258 


2344 


3 


4 2431 


2517 


2603 


2689 


2776 


2861 


2947 


3033 


3119 


3205 


4 


5 8291 


3877 


8463 


3549 


3635 


3721 


3807 


3893 


3979 


4065 


6 


6 4151 


4236 


4822 


4408 


4494 


4579 


4665 


4751 


4837 


4922 


6 


7 5008 


5094 


5179 


5265 


5350 


5436 


5522 


5607 


5693 


6778 


7 


8 5864 


5949 


6035 


6120 


6206 


6291 


6376 


6462 


6547 


6632 


8 


9; 6718 


6803 


6888 


6974 


7059 


7144 


7229 


7315 


7400 


7485 


9 


1 


1 


2 


3 


4 


5 


6 


7 


8 


9 




510 7570 


7655 


7740 


7826 


7911 


7996 


8081 


8166 


8251 


8386 


510 


1 8421 


8506 
9355 


8591 
9440 


8676 
9524 


8761 
9609 


8846 


8931 


9015 


9100 


9185 


1 


2 
3 


9 AIM 


9694 


9779 


9863 


9948 


0033 
0879 


2 
3 


710117 


0202 


0287 


0371 


0456 


0540 


0625 


0710 


0794 


4 0963 


1048 


1132 


1217 


1301 


1885 


1470 


1554 


1639 


1728 


4 


5 1807 


1892 


1976 


2060 


2144 


2229 


2313 


2397 


2481 


2566 


6 


6 2650 


2734 


2818 


2902 


2986 


3070 


3154 


3238 


3323 


3407 


6 


, 7 3491 


3575 


3659 


8742 


3826 


3910 


3994 


4078 


4162 


4246 


7 


1 8 4330 


4414 


4497 


4581 


4665 


4749 


4833 


4916 


5000 


6084 


8 


9 5167 


5251 


5335 


5418 


5502 


5586 


5669 


5753 


5836 


5920 


9 


1 i 


1 


2 


3 


4 


5 


6 


7 


8 


9 




'520 6003 


6087 


6170 


6254 


6337 


6421 


6504 


6588 


6671 


6754 


620 


1 6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


7504 


7587 


1 


2 7671 


7754 


7837 


7920 


8003 


8086 


8169 


8253 


8336 


8419 


2 


3. 8502 

A nooi 


8585 
9414 


8668 
9497 


8751 
9580 


8884 
9663 


8917 


9000 


9083 


9165 


9248 


3 


4 

5 


voox 


9745 


9828 


9911 


9994 


0077 
0903 


4 
5 


720159 


0242 


0325 


0407 


0490 


0573 


0655 


0738 


0821 


6 0986 


1068 


1151 


1233 


1316 


1898 


1481 


1563 


1646 


1728 


6 


7 1811 


1893 


1975 


2058 


2140 


2222 


2305 


2387 


2469 


2552 


7 


8 2634 


2716 


2798 


2881 


2963 


8045 


3127 


3209 


3291 


3374 


8 


9 3456 


3538 


3620 


8702 


3784 


3866 


3948 


4030 


4112 


4194 


9 


I ' 


1 


2 


3 


4 


5 


6 


7 


8 


9 




530 4276 


4358 


4440 


4522 


4604 


4685 


4767 


4849 


4981 


6013 


680 


/ !< 5095 


5176 


5258 


5340 


5422 


5503 


5585 


5667 


5748 


5830 


1 


. 2 5912 
/ 81 6727 


5993 


6075 


6156 


6238 


6320 


6401 


6483 


6564 


6646 


2 


6809 


6890 


6972 


7053 


7134 


7216 


7297 


7379 


7460 


3 


41 7541 


7628 


7704 


7786 


7866 


7948 


8029 


8110 


8191 


8278 


4 


6. 8354 


8485 


8516 


8597 


8678 


8759 


8841 


8922 


9003 


9084 


6 


6 9165 


9246 


9327 


9408 


9489 


9570 


9651 


9732 


9813 


9893 


6 

7 
8 


7 
8 


vv/« 


0055 
0868 


0136 
0944 


0217 
1024 


0298 
1106 


0378 
1186 


0459 
1266 


0540 
1347 


0621 
1428 


0702 
1508 


780782 


9, 1589 


1669 


1750 


1830 


1911 


1991 


2072 


2152 


2233 


2313 


9 


^ 1 
640 28M 


1 


2 


3 


4 


5 


6 


7 


8 


9 




2474 


2555 


2685 


2715 


2796 


2876 


2956 


3037 


3117 


640 


1; 8197 


8278 


8858 


3488 


3518 


3598 


3679 


3759 


3839 


3919 


1 


2 


8999 


4079 


4160 


4240 


4820 


4400 


4480 


4560 


4640 


4720 


2 


' 


4800 


4880 


4960 


5040 


5120 


5199 


5279 


5359 


5439 


5519 


3 


\ * 


5599 


6679 


5759 


5S38 


5918 


5993 


6078 


6157 


6237 


6317 


4 


5 


6897 


6476 


6556 


6635 


6715 


6795 


6874 


6954 


7034 


7113 


6 


\ 2 


7198 


7272 


7852 


7431 


7511 


7590 


7670 


7749 


7829 


7908 


6 


I 


7987 


8067 


8146 


8225 


8305 


8384 


8468 


8543 


8622 


8701 


7 


\i 


8781 
9572 


8860 
9661 


8939 
9731 


9018 
9810 


9097 


9177 


9256 


9386 


9414 


9493 


8 


9889 


9968 


0047 


0126 


0205 1 0284 


9 



10 






TA5L1 IIL 



[ICo. 6999 L. 778079! 



8 



9 N. 



-Mai » -t*!^ -s-sti :^.L. .n*.*. 

* I ! ri. l^i! l^i-ir !!;<* 
1 "! s?!.* i' '. * ? Vf il"r 
3 STiH i>'-§ ir*?i iKT 

♦ *j: •.;=-• 7.^- frii 
■ -is! -.r: ♦44.- 45e* 
f :. -: 5:r.? 52?: «:•> 

" !.•;.: ;:*•?:? f» :: */•■*> 

i" -^r:*-* f~:i f-j.. 5?«* 

> r*:i T**^ r.v~ r*«; 



1 ?iH3 >■*• 



•■?*? 
>'.i* 

>*>! 



'IT 



4 



1»44 

4-f.fl 
*i45 



5 



l<*i4 
2411 
SIM 

4r«2 

«323 

TlOl 

T?:3 

6 
S653 
9427 



0915 
1703 
2439 
3275 
4053 
4S40 
5«21 
6401 
7179 
7955 



8731 
9504 



0994 i 

17S2 

2568 . 

3353 

4136 

4919 

5699 

6479 

7256 

8033 



1073 .560, 



?^^ 



w?A 0971 
1$M 1741 



1048 1125 
1813 18^5 



t^vi 2433 25<.'9 25S6 2663 



1860 
2647 
3431 
4215 
4997 
5777 , 6 
S556 ' 7 
7384 ; 8 
8110 ; 9 



8 ' 9 : ' 

SS08 8885 560 
9582 9659 1 



.»45 0:23 02i>0 0277 0354 0431 



j*^,"*;*? ;:« y.-i -4.. 

4 \ir'r :«^ :*.*.; \s'.'j 

I r:-4* 1115 ir-i r*r* 

« !:•:•? i>i^ *r-.; ^;4T 3:23 3>» 3277 3353 8430 3506 

7 4^<! }.-r.> i'M 3.*: 3 

* 4M? 44i5 4J- : 457* 



1202 
1972 
2740 



3^!^ 3v^ 4<:42 4119 4195 4272 
46J4 473*) 4><>7 4883 4960 5036 



9 


"Hi 


£:i* 


5it5 


5341 


5417 


5494 


5570 


5646 


5722 


6799 


9 




V 


: 


.- 


3 


4 


5 


6 


7 


8 


9 




570 


:.*:: 


:Vi; 


t->:7 


fi-.'VS 


6:j;i 


6256 


6.^2 


6403 


6484 


6560 570 


^ 
A 


e-^- 


■>::2 


?"ji 


6>o4 


6:='40 


7«>16 


7092 


7163 


7244 


7820 




o 


r;-^. 


7472 


754? 


7624 


77C-1 


7775 


7S51 


7927 


8008 


6079 




3 


s::<- 


*i:?^j 


S.^>o 


5&S2 


84^8 


S533 


wk»9 


8685 


8761 


8836 




4 


s^:i 


s^ss 


s--a 


^•139 


y2l4 


92y<i 


9366 


9441 


9517 


9592 




5 


««* 


t743 


9?1> 


9S94 


9k'70 














0799 


0121 
0S75 


0196 
0950 


0272 
1025 


0347 

1101 




! 6: 


6'>422 


•>49S 


0573 


IV849 


0724 




7 


ri:«5 


1251 


132^ 


1402 


1477 


1552 


1627 


1702 


1778 


1858 




: 8 


i**j* 


2.i«>3 


2»»75 


2153 


222S 


23'.'3 


2378 


2453 


2529 


2604 




9 


2679 


2754 


2529 


29i'4 


2978 


3053 


3128 


3203 


3278 


3853 









1 


2 


3 


4 


5 


6 


7 


8 


9 




580 


8428 


3503 


3578 


3653 


8727 


3802 


3877 


8952 


4027 


4101 5S0 


' 1 


4176 


4251 


4326 


4400 


4475 


4550 


4624 


4699 


4774 


4848 


1 


. 2 


4923 


4998 


5*172 


5147 


5221 


5296 


5370 


5445 


5520 


5594 


2 


• 8 


5669 


5743 


5S13 


5892 


5966 


6041 


6115 


6190 


6264 


6838 


8 


4 


6413 


6487 


6562 


6636 


6710 


6785 


6859 


6933 


7007 


7082 


4 


, 5 


7156 


7230 


73rt4 


7379 


7453 


7527 


7601 


7675 


7749 


7828 


5 


i 6 


7898 


7972 


8046 


8120 


8194 


8268 


8342 


8416 


8490 


8564 


6 


7 


8638 


8712 


8786 


8860 


8934 


9008 


9082 


9156 


9230 


9803 


( 


8 


9377 


9451 


9525 


9699 


9673 


9746 


9820 


9894 


9968 


0042 r 


8 


9 770116 


0189 


0263 


0336 


0410 


0484 


0557 


0631 


0705 


0778 


9 




; 


1 


2 


3 


4 


5 


6 


7 


8 


9 




690 


0852 


0926 


0999 


1073 


1146 


1220 


1298 


1867 


1440 


1514 


590 


1 


1587 


1661 


1734 


1808 


1881 


1955 


2028 


2102 


2175 


2248 


1 
•2 


2 


2322 


2395 


2468 


2542 


2615 


2688 


2762 


2885 


2908 


8981 


8 


8055 


3128 


3201 


8274 


3348 


8421 


3494 


8567 


3640 


3718 


S 


4; 


8786, 


3860 


8933 


4006 


4079 


4153 


4225 


4298 


4371 


4444 


4 


6. 


46171 


4590 


4663 


4786 


4809 


4882 


4955 


5028 


5100 


5178 


5 


6 


5246, 


5819 


5892 


5465 


5538 


5610 


5683 


5756 


5829 


5902 


6 


5 


5974 1 


6047 


6120 


6193 


6265 


6838 


6411 


6488 


6556 


66S9 


7, 


6701 


6774 


6846 


6919 


6992 


7064 


7137 


7209 


7888 


7354 


8 


ol 


7427 


7499 


7572 i 7644 


7717 1 7789 


7862 


7984 


8006 1 807» 


9 



11 



Hi. 6000 L. 778161] 




TABLE III. 


[No. 6499 L. 812847 


N. 





1 


2 


8 


4 


5 


6 


7 


8 


9 


N. 
600 


600 


778161 


8224 


8296 


8868 


8441 


8513 


8686 


8668 


8780 


8802 


1 


8874 


8947 


9019 


9091 


9163 


9286 


9308 


9380 


9452 


9624 


1 


2 


9096 


9669 


9741 


9818 


9885 


9957 














0029 
0749 


0101 
0821 


0173 
0893 




8 


780817 0889 


0461 


0538 


0605 


0677 


9246 
0965 


2 
8 


4 


1087 


1109 


1181 


1253 


1324 


1896 


1468 


1640 


1612^ 


1684 


4 


5 


1755 


1827 


1899 


1971 


2042 


2114 


2186 


2268 


2329 


2401 


5 


6 
7 


2473 


2544 


2616 


2688 


2759 


2831 


2902 


2974 


3046 


8117 


6 


8189 8260 8882 


8408 


3475 


3546 


3618 


8689 


3761 


8832 


7 


8 


8904 8975 . 4046 


4118 


4189 


4261 


4382 


4408 


4475 


4646 


8 


9 


4617 4689 1 4760 


4831 


4902 


4974 


6046 


6116 


6187 


6269 


9 




6 1,2 


3 


4 


5 


6 


7 


8 


9 




610 


5880 5401 ! 5472 


5543 


5615 


5686 


6767 


5828 


5899 


5970 


610 


1 


60411 6112 6188 


6254 


6325 


6396 


6467 


6538 


6609 


6680 


1 


2 


67511 6822 ; 6893 


6964 


7035 


7106 


7177 


7248 


7319 


7390 


2 


8 


7460, 7581 7602 


7673 


7744 


7815 


7885 


7966 


8027 


8098 


8 


4 


8168' 


8289 ! 8310 


8381 


8451 


8522 


8693 


8663 


8734 


8804 


4 


6 


8875 


8946 9016 


9087 


9167 


9228 


9299 


9369 


9440 


9610 


5 


6 


9581 


9651 


9722 


9792 


9863 


9933 












0004 
0707 


0074 
0778 


0144 
0848 




7 


790285 


0856 


0426 


0496 


0567 


0637 


0216 
0918 


6 

7 


8 


0988 


1059 


1129 


1199 


1269 


1340 


1410 


1480 


1560 


1620 


8 


9 


1691 


1761 


1881 


1901 


1971 


2041 


2111 


2181 


2262 


2322 


9 







1 


2 


3 


4 


5 


6 


7 


8 


9 




620 


2892 


2462 


2532 


2602 


2672 


2742 


2812 


2882 


2962 


8022 


620 


1 


' 8092 8168 


8231 


3301 


3371 


3441 


3611 


8581 


8661 


3721 


1 


2 


8790 


8860 3980 


4000 


4070 


4139 


4209 


4279 


4349 


4418 


2 


8 


4488 


4558 ! 4627 


4697 


4767 


4836 


4906 


4976 


6045 


6115 


8 


4 


5185 


5254 


5324 


5398 


5468 


5632 


6602 


6672 


6741 


5811 


4 


5 


5880 


5949 


6019 


6088 


6158 


6227 


6297 


6366 


6486 


6506 


5 


6 


6574 


6644 


6713 


6782 


6852 


6921 


6990 


7060 


7129 


7196 


6 


7 


7268 7887 


7406 


7475 


7545 


7614 


7683 


7762 


7821 


7890 


7 


8 


7960 


8029 


8098 


8167 


8286 


8805 


8874 


8443 


8618 


8582 


8 


9 


8651 


8720 


8789 


8858 


8927 


8996 


9065 


9134 


9203 


9272 


9 







1 


2 


3 


4 


5 


6 


7 


8 


9 




10 


9841 


9409 


9478 


9547 


9616 


9686 


9764 


9823 


9892 


9961 


630 
1 


I 


800029 


0098 


0167 


0236 


0306 


0373 


0442 


0611 


0680 


0646 


2 


0717 


0786 


0854 


0923 


0998 


1061 


1129 


1198 


1266 


1885 


2 


\ 


, 1404. 1472 


1541 


1609 


1678 


1747 


1815 


1884 


1952 


2021 


3 




2089 


2158 


2226 


2295 


2363 


2432 


2600 


2668 


2637 


2706 


4 




•774 


2842 


2910 


2979 


3047 


3116 


3184 


3262 


3321 


3389 


5 




8457 


8525 


8594 


8662 


3730 


3798 


8b67 


3935 


4003 


4071 


6 




4189 


4208 


4276 


4844 


4412 


4480 


4548 


4616 


4685 


4753 


7 




4821 


4869 


4967 


5025 


5093 


5161 


5229 


6297 


5365 


5438 


8 




5501 


5569 


5637 


5705 


6778 


6841 


5908 


5976 


6044 


6112 


9 







1 


2 


8 


4 


5 


6 


7 


8 


9 






6180 


6248 


6816 


6384 


6451 


6619 


6587 


6655 


6728 


6790 


640 




6858 


6926 


6994 


7061 


7129 


7197 


7264 


7832 


7400 


7467 


1 




7585 


7603 


7670 


7788 


7806 


7878 


7941 


8008 


8076 


8148 


2 




8211 


8279 


8346 


8414 


8481 


8649 


8616 


8684 


8761 


8818 


3 




8886 


8958 


9021 


9088 


9156 


9223 


9290 


9368 


9425 


9492 


4 




9560 


9627 


9694 


9762 


9829 


9896 


9964 












0031 
0703 


0098 
0770 






>2S8 


0800 


0367 


0434 


0501 


0669 


0636 


0165 
0837 


5 
6 




»904 


0971 


1039 


1106 


1178 


1240 


1807 


1374 


1441 


1508 


7 




575 


1642 


1709 


1776 


1848 


1910 


1977 


2044 


2111 


2178 


8 




M5 


2312 2379 


2445 


2512 


2579 


2646 


2718 


2780 


2847 


9 



1-2 




1359; 1422 
1985 2047 
2609 2672 
3283 3295 
8855 3918 
4477 4589 



0232 
0859 
1485 
2110 
, 2734 
I 8857 
8980 
4601 



13 



No. 

N. 
700 


7000 L. 



845098] 




TABLE III. 




[No. 7499 L, 876003 


1 


2 


3 


4 


5 


6 


7 


8 


9 


N. 


845098 


5160 


^ 5222 


5284 


5346 


5408 


5470 


5532 


5594 


5656 


700 


1 


5718 


5780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 


1 


2 


6337 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 


2 


3 


6955 


7017 


7079 


7141 


7202 


7264 


7826 


7388 


7449 


7511 


8 


4 


7573 


7634 


7696 


7758 


7819 


7881 


7943 


8004 


8066 


8128 


4 


6 


8189 


8251 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


8743 


5 





8805 


8866 


8928 


8989 


9051 


9112 


9174 


9235 


9297 


9358 


6 


7 

8 


9419 


9481 


9542 


9604 


9665 


9726 


9788 
0401 


9849 


9911 


9972 


7 
8 


850083 


0095 


0156 


0217 


0279 


0340 


0462 


0524 


0585 


9 


0646 


0707 


0769 


0830 


0891 


0952 


1014 


1075 


1136 


1197 


9 







1 


2 


3 


4 


6 


6 


7 


8 


9 




710 


1258 


1320 


1381 


1442 


1508 


1564 


1625 


1686 


1747 


1809 


710 


1 


1870 


1981 


1992 


2058 


2114 


2175 


2236 


2297 


2358 


2419 


1 


2 


2480 


2541 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 


2 


8 


8090 


8150 


8211 


3272 


8333 


3394 


3455 


3516 


8577 


3637 


8 


4 


8698 


3759 


3820 


3881 


3941 


4002 


4063 


4124 


4185 


4245 


4 


6 


4306 


4867 


4428 


4488 


4549 


4610 


4670 


4731 


4792 


4852 


5 


6 


4913 


4974 


5034 


5095 


5156 


5216 


5277 


5337 


5898 


5459 


5 


7 


5519 


5580 


5640 


5701 


5761 


5822 


5882 


5943 


6003 


6064 


7 


8 


6124 


6185 


6245 


6306 


6366 


6427 


6487 


6548 


6608 


6668 


8 


9 


6729 


6789 


6850 


6910 


6970 


7081 


7091 


7152 


7212 


7272 


9 


1 





1 


2 


8 


4 


5 


6 


7 


8 


9 




720 


7382 


7893 


7458 


7518 


7574 


7684 


7694 


7755 


7815 


7875 


720 


1 


7935 


7995 


8056 


8116 


8176 


8236 


8297 


8357 


8417 


8477 


1 


2 


8537 


8597 


8657 


8718 


8778 


8838 


8898 


8958 


9018 


9078 


2 


8 


9138 


9198 


9258 


9318 


9379 


9439 


9499 


9559 


9619 


9679 


8 


4 


9739 


9799 


9859 


9918 


9978 


















0038 
0637 


0098 
0697 


0158 
0757 


0218 
0817 


0278 

0877 


4 
5 


5 


860838 


0398 


0458 


0518 


0578 


6 


0987 


0996 


1056 


1116 


1176 


1236 


1295 


1355 


1415 


1475 


6 


7 


1534 


1594 


1654 


1714 


1773 


1833 


1893 


1952 


2012 


2072 


7 


8 


2131 


2191 


2251 


2310 


2370 


2430 


2489 


2549 


2608 


2668 


8 


9 


2728 


2787 


2847 


2906 


2966 


3025 


3085 


3144 


3204 


3268 


9 







1 


2 


3 


4 


5 


6 


7 


8 


9 




780 


8828 


8382 


3442 


3501 


3561 


3620 


3680 


3739 


3799 


3858 


780 


1 


8917 


8977 


4036 


4096 


4155 


4214 


4274 


4833 


4892 


4452 


1 


2 


4511 


4570 


4630 


4689 


4748 


4808 


4867 


4926 


4985 


5045 


2 


8 


6104 


5168 


5222 


5282 


5341 


5400 


5459 


5519 


6578 


6687 


3 


4 


5696 


5755 


5814 


5874 


5933 


5992 


6051 


6110 


6169 


6228 


4 


5 


6287 


6846 


6405 


6465 


6524 


6583 


6642 


6701 


6760 


6819 


6: 


6 


6878 


6937 


6996 


7055 


7114 


7173 


7282 


7291 


7350 


7409 


6 


7 


7467 


7526 


7585 


7644 


7703 


7762 


7821 


7880 


7939 


7998 


7 


8 


8056 


8115 


8174 


8233 


8292 


8350 


8409 


8468 


8527 


8586 


8 


9 


8644 


8708 


8762 


8821 


8879 


8938 


8997 


9056 


9114 


9173 


9 







1 


2 


3 


4 


5 


6 


7 


8 


9 




740 


9282 


9290 


9349 


9408 


9466 


9525 


9584 


9642 


9701 


9760 


740 


1 


9818 


9877 


9935 


9994 






















0053 
0638 


0111 
0696 


0170 
0755 


0228 
0813 


0287 
0872 


0345 
0930 


1 
2 


2 


870404 


0462 


0521 


0579 


8 


0989 


1047 


1106 


1164 


1223 


1281 


1339 


1898 


1456 


1516 


8 


4 


1573 


1631 


1690 


1748 


1806 


1865 


1923 


1981 


2040 


2098 


4 


6 


2156 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 


5 


6 


2739 


2797 


2855 


2913 


2972 


3030 


3088 


8146 


3204 


3262 


6 


7 


8821 


8879 


8487 


8495 


3553 


3611 


3669 


3727 


8785 


8844 


7 


8 


8902 


8960 


4018 


4076 


4134 


4192 


4250 


4308 


4866 


4424 


8 


9 


4482 


4540 


4598 


4656 


4714 


4772 


4880 


4888 


4945 


5008 9| 



14 



Ve. 7M0 L. 875061] TABLB III. 




[Ifo. 7000 L. 0O8O86| 


750 





1 


2 


% 


4 


5 


6 


7 


8 




875061 


5119 


5177 1 6285 


5298 


5851 


5400 


6466 


5524 


5582 ,750 


1 


5640 5698 


5756 581S 


5871 


6980 


5087 


6045 


6108 


6160 1 1 


2 


6218 6276 


6338 1 6891 


6449 


6507 


6564 


6622 


6680 


6787 1 2 


8 


6795 6853 


6910 : 6968 7026 


7088 


7141 


7100 


7266 7814 ' 3 


4 


7871 7429 


7487 7544 ■ 7602 


7659 


7717 


7774 


7882 


7889 , 4i 


5 


7947 8004 


8062 8110 j 8177 


8284 


8202 


8840 


8407 


8464 5' 


6 


8522 8579 


8637 < 8694 > 8752 


8809 


8866 


8024 


8981 


9039 1 6 


7 


9096 9158 


0211 0268 1 0825 


9888 


0440 


0407 


9565 9612 7 


8 


9669 9726 


0784 


9841 


9898 


9956 










1 










0018 


0070 


0127 ' 0185 8 












9 


880242 0299 


0856 0418 


0471 


0528 


0585 


0642 


0600 J 0756 9 




1 


2 8 


4 


6 


6 


7 


8 , • ' 


760 


0814 0871 


0028 0085 


1042 


1099 


1156 


1218 


1271 , 1828 760 


1 


1885 1442 


1499 1556 


1613 


1670 


1727 


1784 


1841 * 1898 1 


2 


1955 2012 


2069 2126 


2183 


2240 


2297 


2854 


2411 1 2468 ' 2 


8 


2525 2581 


2<J38 ; 2695 | 2752 


2809 


2866 


2028 


2980 80S7 3 


4 


8093 8150 


8207 8264 ; 8321 


3377 


8484 


8491 


8548 8605 4 


5 


8661 3718 


8775 8882 i 8888 


8945 


4002 


4059 


4115 41T2 > 5 


6 


4229 4285 


4842 4899 ; 4455 


4512 


4569 


4625 


4682 47S9 6 


7 


4795 4852 


4909 ! 4965 i 5022 


5078 


5135 


5192 


5248 1 5805 7 


8 


5861 5418 


5474 i 5581 5587 


5644 


5700 


5757 


5818 5870 8 





5926 5983 


6039 6096 


6152 


6209 


6265 


6821 


6878 


6434 9 




1 


2 1 8 


4 


5 


6 


7 


8 


t 


770 


6491 6547 


6604 6660 


6716 


6778 


6829 


6885 


6942 


6908 770 


1 


7064 7111 


7167 1 7223 


7280 


7336 


7392 


7449 


7505 


7561 1 


2 


7617 7674 


7730 '■ 7786 


7842 


7898 


7955 


8011 


8067 


8123 2 


8 


8179 8236 


8292 ' 8348 


8404 


8460 


8516 


8578 


8629 


8685 ' 8 


4 


8741 8797 


8858 ; 8909 


8965 


9021 


9077 


9184 


9190 9246 i 


5 


9302 9358 


9414 9470 


9526 


9582 


9638 


9694 


9750 9806 . 5 





9862 9918 

1 


9974 














0086 
0645 


0141 
0700 


0197 
0756 


0253 
0812 


0309 ; 0365 « 
0868 I 0924 7 


7 


890421 0477 


0533 0589 


8 


0980 1035 


1091 1147 


1208 


1259 


1314 


1370 


1426 1482 8 





1537 


1593 


1649 1705 


1760 


1816 


1872 


1928 


1083 


2039 r 







1 


2 i 8 


4 


5 


6 


7 


8 


1 ' 


780 


2095 2150 


2206 i 2262 


2317 


2873 


2429 


2484 


2540 


2505 T80 


1 


2651 2707 


2762 i 2818 


2873 


2929 


2985 


8040 


8006 8151 1, 


2 


3207 8262 


3818 ': 8373 


3429 


8484 


8540 


8505 


3651 


8706. J 


8 


8762 3817 


3878 8928 


8984 


4089 


4094 


4150 


4205 


4861 S 


4 


4316 4371 


4427 ; 4482 


4538 


4593 


4648 


4704 


4759 


4814 < 


5 


4870; 4925 


4980 5086 


5091 


5146 


5201 


5257 


5812 5867- \ 


6 


5423 5478 


5533 i 5588 


5644 


5699 


5754 


5800 


5864 


5920, • 


7 


5975 


6080 


6085 : 6140 


6195 


6251 


6806 


6861 


6416 


6471 ' 


8 


6526 


6581 


6636 


6692 


6747 


6802 


6857 


6912 


6067 


7022 8 





7077 


7182 


7187 


7242 


7297 


7852 


7407 


7462 


7517 


7572 


», 







1 


2 


3 


4 


5 


6 


7 


8 


• 





700 


7627 


7682 


7787 ; 7792 


7847 


7902 


7957 


8012 


8067 


8122 ITW! 
8670 1 


1 


8176 


8281 


8286 


8341 


8396 


8451 


8506 


8561 


8615 


e 


8725 


8780 


8885 


8890 


8944 


8999 


9054 


9109 


9164 


0218 




8 


0278 


0828 


9863 


9487 


9492 


0547 


9602 


0656 


0711 


9766 




4 


0821 


9875 


9980 


9985 
















0089 
0586 


0004 
0640 


0149 
0695 


0808 
0740 


0258 
0804 


0312 
0850 




5 


000867 


0422 


0476 


0581 




a 


0918 


0068 


1022 


1077 


1181 


1186 


1240 


1805 


1840 


1404 




r 


1458 


1518 


1567 


1682 


1676 


1781 


1785 


1840 


1804 


1048 




8 


2008 


2057 


2112 


8166 


8221 


2275 


2829 


2884 


2488 


2198 







2647 


2601 


2655 


8710 


2764 


8818 


8878 


2027 


2081 


8686 


__\ 



15 



N«. 


8000 L. 908090] 




TABLB III. 




[ir«. 8499 L. 9298n! 


N. 





1 


2 


8 


4 


6 


6 


r 


8 


9 


H. 


800 


908090 


8144 


8199 


8258 


8307 


3861 


3416 


8470 


8524 


8578 


800 


1 


8888 


8687 


8741 


8795 


3849 


3904 


3958 


4012 


4066 


4120 


1 


2 


4174 


4229 


4288 


4887 


4391 


4445 


4499 


4663 


4607 


4661 


2 


8 


4716 


4770 


4824 


4678 


4932 


4986 


5040 


5094 


5148 


6202 




4 


5256 


6810 


5864 


5418 


5472 


5526 


5580 


5684 


5688 


5742 




6 


5796 


5850 


5904 


5958 


6012 


6066 


6119 


6173 


6227 


6281 




6 


6885 


.6889 


6448 


6497 


6551 


6604 


6658 


6712 


6766 


6820 




7 


6874 


6927 


6981 


7086 


7089 


7148 


7196 


7250 


7804 


7868 




8 


7411 


7465 


7519 


7578 


7626 


7680 


7784 


7787 


7841 


7895 


8 


9 


7949 


8002 


8056 


8109 


8163 


8217 


8270 


8824 


8878 


8431 


9 







1 


2 


8 


4 


5 


6 


7 


8 


9 




.810 


8485 8589 


8592 


8646 


8699 


8768 


8807 


8860 


8914 


8967 810 


1 


9021 9074 


9128 


9181 


9285 


9289 


9842 


9896 


9449 


9603 ' ll 


2 
8 


9556 9609 


9663 


9716 


9770 


9828 
0858 


98n 


9980 


OOfil 








0037 
0571 


2 
8 


910091 0144 


0197 


0251 


0804 


0411 


0464 0618 


4 


0624 0678 


0781 


0784 


0888 


0891 


0944 


0998 1051 


1104 


4 


' 6 


1158 1211 


1264 


1817 


1871 


1424 


1477 


1630 1584 


1687 


5 


6 


1690 1743 


1797 


1850 


1903 


1956 


2009 


2063 • 2116 


2169 


6 


7 


2222 


2276 


2828 


2881 


2485 


2488 


2541 


2694 2647 


2700 


7 


8 


2758 


2806 


2859 


2918 


2966 


8019 


8072 


8126 3178 


8281 


8 


9 


8284 


8887 


8890 


8443 


8496 


8549 


8602 


8666 8708 


8761 


9 







1 


2 


8 


4 


5 


6 


7 


8 


9 




820 


8814 


8867 


8920 


3973 


4026 


4079 


4132 


4184 


4287 


4290 


820 


1 


4848 4896 


4449 


4502 


4555 


4608 


4660 


4713 


4766 


4819 


1 


2 


4872 


4925 


4977 


5030 


5083 


5186 


5189 


6241 


6294 


5347 


2 


S 


5400 


5458 


5505 


5558 


5611 


5664 


6716 


5769 


5822 


5875 


3 


4 


5927 


5980 


6083 


6085 


6188 


6191 


6248 


6296 


6349 


6401 


4 


6 


6454, 6507 


6559 


6612 


6664 


6717 


6770 


6822 


6875 


6927 


5 


6 


6980 7088 


7085 


7138 


7190 


7248 


7295 


7348 


7400 


7 last 


({ 


7 


7506 7558 


7611 


7663 


7716 


7768 


7820 


7873 


7926 7978 


7 


a 


8080 


8088 


8185 


8188 


8240 


8293 


8846 


8397 


8450 ' 8602 


8 


9 


8555 


8607 


8659 


8712 


8764 


8816 


8869 


8921 


8978 9026 


9 







1 


2 


8 


4 


5 


6 


7 


A 


g 




830 


9078 


9180 


9188 


9285 


9287 


9340 


9392 


9444 9496 


9549 


880 


1 

2 


^601 


9658 


9706 


9758 


9810 


9862 


9914 


9967 






1 
2 


0019 
0541 


0071 
0698 


920128 0176 


0228 


0280 


0332 


0384 


0436 


0489 


S 


0645; 0697 


0749 


0801 


0853 


0906 


0968 


1010 1062 


1114 


8 


4 


11661 1218 


1270 


1322 


1374 


1426 


1478 


1630 


1582 


1634 


4 


6 


1686 1788 


1790 


1842 


1894 


1946 


1998 


2060 


2102 


2154 


5 


6 


2206. 2258 


2810 


2362 


2414 


2466 


2618 


2670 


2622 


2674 


5 


7 


27251 2777 


2829 


2881 


2983 


2985 


8087 


8089 


8140 


8192 


7 
8 


8 


3244: 8296 


8848 


8399 


8451 


8508 


8665 


8607 


8668 


8710 


9 


8762, 8814 


8865 


8917 


3969 


4021 


4072 


4124 


4176 


4228 


9 







1 


2 


8 


4 


5 


6 


7 


8 


9 
4744 




840 


4279 


4881 


4888 


4434 


4486 


4588 


4689 


4641 


4698 


840 


1 


4796 4848 


4899 


4951 


5003 


5054 


5106 


6157 


6209 


5261 


1 
2 


2 


5812 5864 


5415 


5467 


5518 


5570 


5621 


5678 5725 


5776 


8 


5%2» 5879 


5981 


5982 


6084 


6085 


6187 


6188 6289 


6291 


8 

4 
5 
6 
7 
8 
9 


4 


6842 6894 


6445 


6497 


6548 


6600 


6651 


6702 ; 6764 


* 6805 


6 


6857 


6908 


6959 


7011 


7062 


7114 


7166 


7216 i 7268 


7819 
7832 


8 


7870 


7422 


7478 


7524 


7576 


7627 


7678 


7780 1 7781 


7 


7888 


7985 


7986 


8087 


8088 


8140 


8191 


8242 


8298 


8845 


8 


8896 


8447 


8498 


8549 


8601 


8652 


8708 


8754 


8805 


8857 


9 


8908 


8959 


9010 


9061 


9112 


9163 


9S15 


9266 


19817 


9868 



16 



,No. S5iiO L. 929419] 



TABLE III. 



[No. 8999 L. 954194 




1240 , 1289 ?i 

1726 ' 1776 { 

2211 . 2259 5 

2696 I 2744 < 

3180 I 3228 : , 

866S i 8711 ^i 

4146 I 4194 I ' 



17 



5«. 9000 L. W42i«\ 




TABLB XXL 




Pf 0. 9499 L. 977678 


K. 





1 


2 


8 


4 


5 


6 


7 


8 


9 


N. 


990 


984248 


4291 


4339 


4387 


4436 


4484 


4582 


4580 


4628 


4677 


900 


1 


4726 


4778 


4821 


4869 


4918 


4966 


5014 


5062 


5110 


5168 


1 


2 


8207 


5288 


8308 


5352 


5399 


5447 


5495 


5543 


5692 


5640 


2 


3 


8888 


5736 


5784 


5882 


5880 


5928 


5976 


6024 


6072 


6120 


3 


4 


8188 


6216 


6265 


6313 


6861 


6409 


6457 


6505 


6653 


6601 


4 


5 


8649 


6697 


6745 


6798 


6840 


6888 


6936 


6984 


7032 


7080 


5 


6 


7128 


7176 


7224 


7272 


7320 


7868 


7416 


7464 


7612 


7559 


6 


7 


7807 


7688 


7703 


7751 


7799 


7847 


7894 


7942 


7990 


8038 


7 


8 


8086 


8184 


8181 


8229 


8277 


8825 


8373 


8421 


8468 


8516 


Sj 


9 


8864 


8612 


8659 


8707 


8755 


8803 


8850 


8898 


8946 


8994 


9 







1 


2 


3 


4 


6 


6 


7 


8 


9 




910 


9041 


9089 


9137 


9185 


9282 


9280 


9328 


9375 


9423 


9471 


910 


1 


9818 


9586 


9614 


9861 


9709 


9757 


9804 


9852 


9900 


9947 


1 


^i 


•fifl/k 






















wvo 


0042 
0518 


0090 
0566 


0138 

0613 


0185 
0661 


•233 

0709 


0281 
0766 


0328 
0804 


6376 
0851 


0428 
0899 


2 
3 


8 


980471 


4 


0946 


0994 


1041 


1089 


1136 


1184 


1281 


1279 


1326 


1874 


4 


6 


1421 


1489 


1816 


1863 


1611 


1658 


1706 


1753 


1801 


1848 


5 





1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


2322 


6 


7 


2889 


2417 


2464 


2511 


2559 


2606 


2663 


2701 


2748 


2796 


7 


8 


2848 


2890 


2987 


2988 


3032 


8079 


8126 


3174 


8221 


3268 


8 


9 


8816 


3863 


3410 


3457 


8504 


8552 


8599 


3646 


3693 


3741 


9 







1 


2 


3 


4 


5 


6 


7 


8 


9 




ftIO 


8788 


3838 


3882 


3929 


3977 


4024 


4071 


4118 


4165 


4212 


920 


1 


4260 


4807 


4854 


4401 


4448 


4495 


4542 


4690 


4637 


4684 


1 


2 


4781 


4778 


4825 


4872 


4919 


4966 


5018 


5060 


5108 


6155 


2 


8 


8202 


5249 


5298 


5343 


5890 


5487 


6484 


5631 


6678 


5625 


8 


4 


8872 


5719 


5766 


5813 


5860 


6907 


5954 


6001 


6048 


6096 


4 


5 


8142 


8189 


6286 


6283 


6829 


6376 


6428 


6470 


6617 


6664 


6 


8 


6611 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


6986 


7083 


6 


7 


7080 


7127 


7178 


7220 


7267 


7314 


7361 


7408 


7464 


7601 


7 


8 


7848 


7595 


7642 


7688 


7736 


7782 


7829 


7876 


7922 


7969 


8 


9 


8016 


8062 


8109 


8156 


8203 


8249 


8296 


8348 


8389 


8486 


9 







1 


2 


8 


4 


6 


6 


7 


8 


9 




980 


8483 


8580 


8576 


8628 


8670 


8716 


8763 


8810 


8866 


8903 


930 


1 


8980 


8998 


9048 


9090 


9186 


9183 


9229 


9276 


9823 


9869 


1 


2 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9836 


2 


8 


9882 


9928 


9975 


















0021 
0486 


0068 
0588 


0114 
0579 


0161 
0626 


0207 
0672 


0254 
0719 


0800 
0765 


3 

4 


4 


970847 


0398 


0440 


6 


0812 


0858 


0904 


0951 


0097 


1044 


1090 


1187 


1188 


1229 


5 


6 


12T6 


1822 


1869 


1415 


1461 


1608 


1564 


1601 


1647 


1698 


6 


7 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


2110 


2157 


7 


( 


2203 


2249 


2295 


2842 


2888 


2434 


2481 


2527 


2678 


2619 


8 


fl 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


8085 


8082 


9 







1 


2 


8 


4 


5 


6 


7 


8 


9 




I4C 


8128 


3174 


8220 


3266 


3813 


8869 


3405 


3451 


8497 


8543 


940 


1 


8880 


3636 


8682 


3728 


3774 


8820 


8866 


3913 


8969 


4005 


1 


3 


4081 


4097 


4148 


4189 


4286 


4281 


4327 


4374 


4420 


4466 


2 


2 


4812 


4858 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 


3 


4 


4972 


8018 


8064 


5110 


5156 


5202 


5248 


5294 


5840 


5886 


4 


e 


8482 


8478 


8524 


5570 


5616 


5662 


5707 


5753 


5799 


5846 


6 


fl 


8891 


8987 


8988 


6029 


6075 


6121 


6167 


6212 


6258 


6304 


6 


7 


8880 


6396 


6442 


6488 


6538 


6579 


6625 


6671 


6717 


6763 


7 


8 


MOO 


8854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7920 


8 


9 


7886 


7812 


7858 


7408 


7449 


7495 


7641 


7586 


7632 


7678 


9 



18 



No. 9600 L. 977724] 


TABLE III. 




[No. 9999 L. 999987! 


N. 





1 


2 


8 


4 


6 


6 


7 


8 


9 N. 


050 977724 


7769 


7816 


7861 


7906 


7962 


7998 


8043 


8089 


8185 950 


1 


8181 


8226 


8272 


8817 


8863 


8409 


8464 


8600 


8546 


8591 1 


2 


8637 


8683 


8728 


8774 8819 


8865 


8911 


8956 


9002 


9047 S 


3 


9093 


9138 


9184 


9230 9276 


9321 


9366 


9412 


9457 


9503 8 


4 
5 


9548 


9594 


9639 


9686 1 9730 


9776 


9821 


9867 


9912 


9958 4 


980003 


0049 


0094 


0140 ' 0186 


0281 


0276 


0822 


0867 


0412 6' 


6 


0458 


0503 


0549 


0594 


0640 


0685 


0730 


0776 


0821 


0867 6, 


7 


0912 


0957 


1008 


1048 


1093 


1189 


1184 


1229 


1275 


1820 7l 


8 


1366 


1411 


1456 


1501 


1647 


1692 


1637 


1688 


1728 


1778 1 8 


9 


1819 


1864 


1909 


1954 1 2000 

1 


2045 


2090 


2185 


2181 


2226 9 

1 







1 


2 


8 


4 


6 


6 


7 


8 


• 1 


960 


22n 


2816 


2362 


2407 


2462 


2497 


2643 


2588 


2688 


2678 |960| 
8180 ll 


1 


2728 


2769 


2814 


2859 


2904 


2949 


2994 


8040 ■ 8085 


2 


8175 


3220 


8266 


8310 


8856 


8401 


8446 


8491 ' 3586 


8581 1 Si 


8 


8626 


3671 


8716 


3762 i 8807 


8852 


8897 


8942 , 8987 


4082 S- 


4 


4077 


4122 


4167 


4212 . 4257 


4802 


4847 


4892 1 4487 


4482 4i 


5 


4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 4887 


4982 6; 


6 


4977 


6022 


6067 


6112 


6167 


6202 


6247 


6292 5837 


5882 6' 


7 


5426 


6471 


6516 


6561 


6606 


6651 


6696 


6741 5786 


5880 1 7i 


8 


6876 


5920 


6965 


6010 


6056 


6100 


6144 


6189 1 6284 


6279 1 Si 


9 


6324 


6369 


6418 


6458 


6593 


6548 


6598 


6687 6682 


6727 9 







1 


2 


8 


4 


6 


6 


7 


8 


» 


970 


6772 


6817 


6861 


6906 1 6951 


6996 


7040 


7085 


7180 


7176 970 


1 


7219 


7264 


7309 


7863 ! 7398 


7448 


7488 


7682 


7577 


7622 1 


2 


7666 


7711 


7766 


7800 


7846 


7890 


7984 


7979 


8024 


8068 i 


8 


8113 


8157 


8202 


8247 


8291 


8386 


8881 


8425 , 8470 


8514 i 


4 


8559 


8604 


8648 


8698 


8737 


8782 


8826 


8871 1 8916 


8960 4 


6 


9005 


9049 


9094 


9138 ' 9188 


9227 


9272 


9816 9361 


9405 5 


6 


9450 


9494 


9639 


9688 9628 


9672 


9717 


9761 9806 


9850 « 


7 


9895 


9939 


9983 












0028 0072 
0472 1 0616 


0117 
0561 


0161 
0605 


0206 1 0250 
0650 0694 


0294 r 

0788 8 


8 


990339 


0388 


0428 


9 


0783 


0827 


0871 


0916 0960 

1 


1004 


1049 


1098 


1187 


1182 9 







1 


2 


8 


4 


6 


6 


7 


8 


9 


980 


1226 


1270 


1316 


1869 


1408 


1448 


1492 


1586 


1580 


1625 980 


1 


1669 


1718 


1768 


1802 


1846 


1890 


1935 


1979 


2028 


2067 1 


2 


2111 


2166 


2200 


2244 


2288 


2888 


2877 


2421 


2465 


2509 i 


8 


2564 


2598 


2642 


2686 2730 


2774 


2819 


2868 


2907 


2961 8 


4 


2995 


3039 


3088 


3127 8172 


8216 


8260 


8804 


8848 


8892 j 4 


5 


8436 


8480 


8524 


8668 8618 


8667 


8701 


8745 


8789 


88331 6 


6 


8877 


8921 


8966 


4009 4068 


4097 


4141 


4185 


4229 


4878 1 « 


7 


4817 


4361 


4406 


4449 4498 


4687 


4681 


4625 


4669 


4718; 7 


8 


4767 


4801 


4846 


4889 1 4938 


4977 


6021 


5065 


5108 


5158 1 8 


9 


6196 


6240 


6284 


6328 ! 6372 


6416 


6460 


5504 


5547 


5591 9 







1 


2 


1 
8 1 4 


6 


6 


7 


8 


9 


990 


6685 


6679 


6728 


6767 ! 6811 


6864 


6898 


5942 


5986 


6080 990 


1 


6074 


6117 


6161 


6206 


6249 


6298 


6887 


6880 


6424 


6468 1 


2 


6612 


6566 


6699 


6648 


6687 


6781 


6774 


6818 


6862 


6906 S 


8 


6949 


6998 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7848 8 


4 


7886 


7480 


7474 


7617 


7661 


7606 


7648 


7692 


7786 


7779 4 


6 


7823 


7867 


7910 


7964 


7998 


8041 


8086 


8129 


8172 


8216 f 


6 


8269 


8808 


8847 


8890 


8484 


8477 


8621 


8564 


8608 


8652 6 


7 


8696 


8789 


8782 


8826 


8869 


8918 


8966 


9000 


9048 


9087 T 


8 


9180 


9174 


9218 


9261 


9806 


9848 


9892 


9485 


9479 


9528 8 


9 


9666 


! 9609 


9662 


9696 


9789 


9788 


9826 


9870 


9918 


9957 9 



10 



Deo. 



TABLB lY.— Loo. Snris, no. 



Sine 




86 
87 
88 
89 
40 
41 



48 086965 

48 8.097188 

44 107167 

"' 116926 



8 866816 ;--" 

9 417968 lllll 

10 463725 lll^ 

11 605118 2S2? 

12 542906 ^"^^^ 
18 7.577668 ^J^JJ 
14 609858 f3«*J 
16 689816 lll^l 

16 667846 ^Jj* 

17 694178 «;5J 

18 7189971 ll^ll 

19 7.742477^ ^\^ 

20 764754! llltL 
21 

28 825451 |2175 

24 843984 K 

25 7.861662 „„«» 

26 878695. *?|?; 

27 895085! -Jj^i 

28 910879 JSoo 

29 926119 *JJS 

^ .!tr«o ^^^ 

81 7.955082 ^ 

82 968870 ^g 
188 982288' *J*JS 
!84 995198 *1J2? 

85 8.007787j IgSSJ 

86 020021! ?2!?i 



S.081919 
043501 
054781 



19802 
18801 



076500 ii?I* 



076500 
086965 

,^ I N "'"— ■ *- 

45 J.XVV2V 

46 126471 

47 185810 

48 144958 



50 162681 

171280 j:^ 



49 8.158907 

51 1 # xxov 

179718 ^I^SS 



» 187985 J^XX 

54 196102 \f^ 
K 8.204070 laJT; 

56 211895 J22JJ 

57 219581 }|!iS 

58 227184 {^4 
19 284557 igS 
«J 2 41855 "^^ 



17441 
17081 
16689 
16265 
15908 
15566 
15288 
14924 
14622 
14888 



COMC. 



Infinite. 

13.586274 
285244 
059158 

12.984214 
837304 
758128 

12.691176 
633184 
582082 
536275 
494882 
457094 

12.422382 
890147 
860184 
332155 
805827 
281008 

12.257523 
235246 
214057 
193854 
174549 
156066 

12.188338 
121805 
104915 
089121 
078881 
059158 

12.044918 
081180 
017767 
004802 

11.992218 
979979 

11.968081 
956499 
945219 
984224 
928500 
918035 

11.902817 
892883 
888074 
878529 
864190 
855047 

11.846098 
887819 
828720 
820287 
812015 
808898 

11.795980 
788105 
780419 
772866 
765448 
758145 



Tang. 



D. 



0.000000 

6.463726 
764756 
940847 

7.066786 
162696 
241878 

7.808825 
866817 
417970 
463727 
505120 
542909 

7.577672 
609857 
639820 
667849 
694179 
719003 

7.742484 
764761 
785951 
806155 
825460 
843944 

7.861674 
878708 
895099 
910894 
926134 
940858 

7.955100 
968889 
982253 
995219 

8.007809 
020045 

8.031945 
048527 
054809 
065806 
076581 
086997 

8.097217 
107202 
116963 
126510 
185851 
144996 

8.153952 
162727 
171328 
179763 
188036 
196156 

8.204126 
211953 
219641 
227195 
284621 
241921 



601717 
293485 
208281 
161517 
131969 
111578 
96653 
85254 
76263 
68988 
62981 
57938 
58642 
49989 
46715 
43882 
41373 
39136 
37128 
85316 
33673 
32176, 
308071 
29549 
28890 
27318 



Cotang. 



Secant Ootang. 



254011 
245401 
28785 
22982 
22276 
21610 
20983 
20392 



Infinite. 

13.636274 
285244 
069153 

12.934214 
837304 
768122 

12.691176 
633188 
582030 
536278 
494880 
467091 

12.422328 
390143 
860180 
832161 
805821 
280997 

12.257516 
235239 
214049 
193845 
174640 
156066 

12.138326 
121292 
104901 
089106 
073866 
069142 

12.044900 
081111 
017747 
004781 

11.992191 
979966 



19888' 

198051 ^^•^^^^^'^ 
J?r!!xl 966473 



18808 
I8327I 
17875 
17444 
17084 



945191 
934194 
923469 
913003 



16642; ll-»02783 

16268, 
15912 
15568 
16241 
149271 
14625 
148861 
14067| 
13790, 
18532, 
182841 
13044 
12814J 
12590'| 



12876 
12168 



D. 



892798 
883037 
873490 
864149 
865004 

11.846048 
887273 
828672 
820287 
811964 
803844 

11.795874 
788047 
780359 
772805 
766879 
758079 



Secant D.l Cosine 



10.000000 





1 

10.000001 

1 
1 

2 
2 
8 

10.000003 
4 
4 
6 
5 
6 

10.000007 

7 

8 

9 

10 

11 

10.000011 

12 

13 

14 

15 

17 

10.000018 

19 

20 

21 

28 

24 

10.000025 

27 

28 

29 

81 

82 

10.000084 

36 

87 

89 

41 

42 

10.000044 

46 

48 

50 

52 

54 

10.000056 

58 

60 

62 

64 

66 



Tang. 



CoMC. 



10.000000 











9.999999 

9.999999 

9 

9 

8 

8 

7 

9.999997 

6 

6 

5 

5 

4 

9.999993 

3 

2 

1 



89 

9.999989 

88 

87 

86 

85 

83 

9.999982 

81 

80 

79 

77 

76 

9.999975 

78 

72 

71 

69 

68 

9.999966 

64 

68 

61 

59 

58 

9.999956 

54 

52 

50 

48 

46 

9.999944 

42 

40 

88 

86 

84 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
86 
85 
84 
38 
82 
81 
30 
29 
28 
27 
26 
25 
24 
28 
22 
21 
20 
19 
18 
17 
16 
15 
14 
18 
12 
11 
10 
9 
8 
7 
6 
5 
4 
8 
2 
1 




Sine 

89 Dto. 



20 



1 Deg. 



TABLE lY.— Loo. 8iffB, im 



Sin* 



Coflec. 



Tang. 



Cotang. 



,8.241855 

1 I 41*033 

2 , 56«>y4 
63042 
69SS1 
7t>614 
83243 

7 . 8.289773 

8 96207 

9 8.302546 
10 . 08794 
U. 14954 
12 , 21027 
18 8.327016 
14 , 32924 



15. 
16 
17. 
18: 



38753 
44504 
501S1 
55783 



19 8.361315 



20. 

21 

22! 

23 j 

24 I 



66777 
72171 
774in» 
82762 
87962 



25 8.393101 

26 98179 
27.8.403199 



28 
29 
80 



08161 
13068 
17919; 



32 



31 ; 8.422717| 
274621 
32156: 
36800 
41394' 
45941: 



34 I 

35 I 
36 t ,^..., 

87 : 8.450440 

88 i 54893 



39! 
401 

42 I 



59801 
63665] 
67985; 
72263, 



44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
,67 
i58 
59 
60 



80693 
84848 
88963 
93040 
97078 

8.501080 
05045 
08974 
12867 
16726 
20551 

8.524843 
28102 
81828 
85528 
89186 
42819 



11963 
1176S 
115S0 
1139S 
11221 
11050 
10883 
10723 
10565 
10413 
H.»266 
10122 
9982 
9S47 
9714 
9586 
9460 
9338 
9219 
9103 
8990 
8880 ; 
8772 i 
8667 j 
8564 
8464 
8366 
8271 
81771 
8086, 
7996; 
7909 ; 
7823 ! 
7740 
7657 
7577 
7499 
7422 
7846 
7273 
7200 
7129 
7060 
6991 
6924 
6859 
6794 
6781 
6669 
6608 
6548 
6489 
6482 
6375 
6319 
6264 
6211 
6158 
6106 
6055 



11.758145 8.241921 
50967 49102 
43906 
86958 
30119 
23386 
16757 



56165 
63115 
69956 
76691 
83323 



11.710227 
03793 

11.697454 
91206 
85046 
78973 

11.672984 
67076 
61247 
55496 
49819 
44217 

11.638685 
33228 
27829 
22501 
17238 
12033 

11.606899 
01821 

11.596801 
91839 
86932 
82081, 

11.577283' 
72538 
67844 
63200 
58606^ 
54059| 

11.549560, 
4510' 
40699 
86335 
82015 
2773 

11.523502 
1930! 
15152 
11037 
06960 
02922 

11.498920 
94955 
91026 
87133 
83274 
79449 

11.47565' 
71898 
68172 
64477 
60814 
57181 



8.289856 

96292 
8.302634' 

08884 

15046 

21122^ 
8.827114' 

33025 

38856 

44610. 

50289 

55895. 
8.861430' 

66895- 

722921 

77622' 

82889! 

88092; 
8.893234- 

98315; 
8.4033381 

08304! 

13213J 

18068{ 
8.422869! 

27618! 

82315 

86962 

41560 :: 



46110 

8.450613 
55070 
59481! 
638491 
68172 
72454 

8.476693* 
80892 
85050j 
89170 
03250 
97293 

8.501208 
05267 
09200 
13098 
16961 
20790 

8.524586 
88349 
82080 
85779 
89447 
43084 



11967 
11772 
11584' 
11402 
11225' 
11054' 
10887 ■ 
10726 
10570 
10418 
10270 
10126' 
9987- 
9851- 
9719' 
9590 
9465 
9343! 
9224 
9108 
8995 
8885 
8777' 
8672' 
8570 
8470, 
8371' 
8276 ' 
8182! 
8091 1 
8002' 
7914 
7828 
7745 
7663 
7583 
r505 



Seomt ID.' Counc 



I 



7428 
7352 
7279 
7206 
7135 
7066 
6998 
6931 
6865 
6801 
6738 
6676 
6615 
6555 
6496 
6439 
6882 
6826 
6272 
6218 
6165 
6118 
6062 



11.758079 
5089d[ 
43885| 
86885, 
80044 
38309 
166771 

11.710144; 
08708 I 

11.697366| 
91116 
84954 
78878- 

11.672886 
66975 
61144 
65390 
49711 
44105 

11.688570 
83105 
27708 
82878 
17111 
11908 

11.606766 
01685; 

11.596662; 
91696 
867871 
81982 

11.577131! 
72382; 
67685 
63038| 
58440 
53890 

11.549387 
44930| 
40519' 
86151! 
818281 
27546| 

11.523807; 
19108' 
14950| 
10830| 
06750| 
02707 



10.000066 

68- 
71 i 
78! 

75! 

80' 
10.000082* 
85 
87? 
90! 

9s; 
95; 

10.000098 
10.000101; c 

^2 5 
10.000115! , 

8l'^ 



24 
271 
80 
10.000133 
Wi 
89 
42 

49j 
10.000152 
66 
69i 
62 
66| 
69 



5 
5 

f! 
ilj 

;i5; 
,!«' 

6 
6 
6 
6 
6 
6 



11.498702! 
94738 
90800 
86902 
88089 
79210 

11.475414 
71651 
67920 
64221 
60558 
56916 



10.000178; « 
77|« 
80| 
84' 
88 
91 

10.000195 
99 

10.000208 
07 
10 

10.000218' 
22; 
26 
81 
85 
89 

10.000843 
47 
88 
68 
80 
88 



9.999984 60. 

82 591 

29 58 

27 57' 

85 56 

82 55 

ao;54 

9.999918 > 58 , 
15 52 
18.51 
10 50 
07 49 
05 48 

9.999902; 47 

9.999899 1 46 
97i45 
94 .44 
91148 
88:42 

9.999885 1 41 
82.40 
79! 89 
76 1 88 
78.87: 
70;8l! 

9.999867 185 1 
84.84,- 
6ljl9< 
68 8i| 
54 '81 1 
51 



9.999848 
44 

41 



tsiu' 

84j» 

81 24 1 

9.999827 21: 

90|»| 

16|Mi 

i«'w| 

9.999805,17, 

01 IK 

9.999797 j IS, 

»8!H. 
90!IS 
86-lS' 

9.99978S;n 

78:W 

74I 9 

69 

65 

61 
9J90767 

58 

46 

44 

46 

85 



Cotiiie D. 



Secant 



Cotang. 



T^. 



88 Did. 



91 



a Deg. 



D. 



' j Sii»« I 

4 670M llf? 

5 60640 Sis 

6 6»W9|^JJ^ 
7-&M74S1; ^^. 

10 

11 
Ifi 



GOMC. 



TABLE IV^Loo. Sins, xto. 
Cotang. 



Tang. 



D. 



Secant D. Codne 



9.999786 60 
81 I 69 
26 68 
22i67 
17 '66 

18 66 
08 64 

9.999704 ' 63 
9.999699 62 
94 61 
89 60 
86 49 
80 48 

9.999676 47 
70 46 
66,46 
60 44 
66 43 
60,42 

9.999646 41 

40 40 
36 89 
29 88 
24 87 

19 86 
9.999614 85 

08.84 
03 83 

9.999697 82 
92 81 
86 80 

9.999681 29 
75,548 
70 27 
64 I 26 
68 26 
63|24 

9.999647 I 28 

41 I 22 
86; 21 
29 120 

19 
18 



708W Sao 

7*214 IS? 

77666 ^l 

80892 "** 

12 84198, ^2 

13 8.687469' ^^.^ 

14 907211 "J; 

16 98948) ?;;: 

16 971621 ^l 

17 8.600882 ^V 

18 08489; l^ 

ao 09784i ^JJ5 

12828 "J* 



M 
26 
26 
27 
28 
29 
90 



84 
88 



6076 
6041 
6006 
4972 



18987 

21962 

26 &824066 

SS854 J"* 

86776 AQ9Q 
80 89680 ^^ 

82 46428 %Vi 
48274 J?g 

"^<^ 4682 

87 8.668476 .^« 

88 62280 VSi 

40 67689 1^ 

41 70898 JJSJ 

42 78080 J*S 
48 &676761 ..q. 
^ 78406 !!^ 
^ ^IM' 4870 

46 88666 !:;? 

47 86272 :J:J 

48 88868 Z^l 
48 8.681488 .our 

60 98998 VSit 

61 96648 Jf;^ 

62 99078 Jfjl 
68 8.701689 ZVZ 
64 04090 4!?? 
66 8.706677 
66 00049 



67 
68 

68 



"09049 JJIJ 

11607 JJ5i 

18962 AAKt 

16888 J^ 

18800 *^^ 



r©: 



11.467181 
68678 
60006 
46461 
42946 
89460 
86001 

11.482669 
291641 
267861 
22484 
191081 
16807 

11.412681 ' 
092791 
06062) 
02848! 

11.899668 
96611, 

11.898877 
90266 
871771 
841091 
810681 
78088' 

11.875086 
72062: 



66146 
68224; 
60820 

11J67487 
64672 
61726 
48898 
46089 
48298 

11.840526 
87770! 
86082! 
82811 
29607 
26920! 

11J24249 
21696i 
189671 
16886 
18728 
11187 

11J08662 
06002 
08467 
00927 

11.298411 
96910 

11.298428 
90961 
884981 
86048 
88617 
81200 



8.648084 
46691 
60268 
63817 
67836 
60828 
64291! 

8.667727: 
711371 
74620' 
77877; 
81208 
84614 

8.687796' 
91061 
94288 
97492 

8.600677: 
03889! 

8.606978 
10094; 
18189| 
162621 
19313' 
22843; 

8.626862: 
28340, 
81308> 
84266 
871841 
40093 

8US42982 
46868 
48704 
61637 
64862 
67149 

8.659928 
62689 
66483 
68160, 
70870 
73668 

8.676289 
78900; 
81644! 
84172 
86784 
89381 J 

8.691963 
94629, 
97081! 
99617i 

8.702189; 
04646 

8.707140! 
09618; 
12083 
14634 
16972 
19896 



Secant i Cotang. 



6012 
6962 
6914 
6866 
6819 
6773 
6727 
6682 
6638 
6695 
6652 
6510 
6468 
6427 
6387 
6847 
6808 
6270 
6232 
6194 
6158 
6121 
6085 
6060 
6016 
4981 
4947 
4913 
4880 
4848 
4816 
4784 
4768 
4722 
4691 
4661 
4631 
4602 
4673 
4644 
4617 
4488 
4461 
4484 
4407 
4380 
4864 
4828 
4803 
4277 
4262 
4228 
4203 
4179 
4165 
4182 
4108 
4086 
4062 
4040 



D. 



11.456916 
58309 
49732 
46183 
42664 
89172' 
85709' 

11.482278 
28868 
25480: 
22123 
18792 
15486', 

11.412206 
08949 
057171 
02508 

11.899828! 
96161 

11.898022 
89906! 
868111 
837381 
806871 
77657. 

11.374648 
71660 
686921 
65744 
62816 
69907 

11.367018 
64147, 
61296 
48463 
46648; 
42851 ! 

11.340072 
3781 Ij 
34667] 
31840' 
291301 
264371 

11.828761 
21100: 
18466: 
15828 
13216 
10619 

11.808087 
05471 
02919 
00888 

11.297861 
95364 

11.292860 
90882 
87917 
86466 
88028 
80604 



10.000265 
69 
74 
78 
83 
87 
92 
10.000296 
10.000801 



U 
15 
20 
10.000326 
80 
85 



8l 



40 gl 

46,8' 
60 gl 

10.000356 9 



60 
66 
71 
76 
81 

10.000386' 
92 
97 

10.000408 
08 
14 

10.000419 

80 
86 



l^io 

47 10 
10.000453 iQ 



69 
65 
71 
76 



82 10 
10.000488 ^0 



94 



10 



1O.OOO6OO,J0 



Tang. 



"Ssi 



10 



13 

1»,10I 
10.000625. 0' 

81 /" 

87i 

44 

60 

67,: 
10.000568= 

691 



76 

82;: 
89: 
96i 



COMC. Il>. 



24 
18 



9.999612 17 



06 
00 

9.999493 
87 
81 

9.999476 
69 
68 
66 
60 
43 

9.999487 
81 
24 
18 

• 11 
04 



16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 





Sine 

"87 Deo. 



dii 



3 Deo. 



TABLB IV.— Loo. Suia, nc. 



Sine I D. 



i 8.718800 



II 

2 

8 

4 

5 

6 

7 

8 

9 

10 

11 

12 

18 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

80 

81 

32 

88 

84 

85 

86 

37 

38 

89 

40 

41 

42 

48 

44 

45 

46 

47 

48 

49 

50 

51 

52 

58 

54 

55 

56 

57 

58 

59 

60 



21204 

23595 

25972 

28837 

80688 

83027 
8.785354 

87667 

89969 

42259 

44536 

46802 
8.749055 

5129' 

53528 

55747, „^.„ 

57965' 22i? 



4006 
3984 
3962 
3941 
3919 
8898 

I 3877 
8857 
3836 
3816 
3796 
8776 
3756 
3737 
3717 

! 3698 



Coiec. I Tang. 



11.281200 8.719896 
78796': 21806 
76405 
74028 
71663 
69312 
669731 



D. ! Cotang. | Seont iP. 



24204 
26588 
28959 
81817 
83663 



11.264646 8.785996 



60151 

8.7623371 

64511 

66675 

68828 

70970 

73101 

8.775228 

77333 

79434 

81524 

83605 

85675 

8.787736 

89787 

91828 

93859 

95881 

97894 

8.799897 

8.801892 

08876 

05852 

07819 

09777 

8.811726 

18667 

15599 

17522 

19436 

21343 

8.828240 

25180 

27011 

28884 

30749 

82607 

8.884456 

86297 

88130 

89956 

.41774 

48585 



3661 

3642 

3624 

8606 

3588 

8570 

3553 

3535 

3518 

3501 

8484 

8467 

3451 

8434 

3418 

3402 

8386 

3370 

3354 

3339 

3323 

8308 

3293 

8278 

3263 

8249 

8234 

3219 

3205 

3191 

3177 

3163 

3149 

3135 

3122 

3108 

8095 

8082 

3069 

3056 

3043 

3030 

3017 



Coiine 



62383 
60031 
57741 
55464 
53198 
11.250945 
48703 
46472 
44253 
42045 
39849 
11.237663 
85489 
33325 
31172 
29030 
26899 
11.224777 
22667 
20566 
18476 
16895 
14325 
11.212264 
10218 
08172 
06141 
04119 
02106 
11.200103 
11.198108 
96124 
94148 
92181 
90228 
11.188274 
86333 
84401 
82478 
80564 
78657 
11.176760 
74870 
72989 
71116 
69251 
67393 
11.165544 
68703 
61870 
60044 
58226 
56415 



88317 
40626 
42922 
4520 
47479 
8.749740 
51989 
54227 
56453 
58668 
60872 
8.763065 
65246 
67417 
69578 
7172' 
73866 
8.775995 
78114 
80222 
82820 
84408 
86486 
8.788554 
90613 
92662 
94701 
96731 
98752 
8.800763 
02766 
04758 
06742 
0871 
10683 
8.812641 
14589 
16529 
18461 
20384 
22298 
8.824205 
26103 
27992 
29874 
81748 
83613 
8.885471 
87821 
89163 
40998 
42825 
44644 



4017 

8995 

8974 

8952 

8930 

3909 

8889 

8868 

8848 

3827 

3807 

8787 

3768 

3749 

3729 

8710 

3692 

8673 

3655 

3636 

8618 

8600 

3583 

3665 

3548 

8531 

8514 

3497 

8480 

8464 

8447 

8431 

3415 

3899 

3383 

3368 

8352 

8337 

3822 

8807 

3292 

8277 

3262 

8248 

3233 

3219 

3205 

3191 

8177 

8163 

8150 

3186 

3123 

3108 

8096 

8088 

3070 

3057 

3045 

8082 



Conne 



11.280604 10.000596,, 9.999404 
78194; 10.000602 :f 9.999898 
75796- 09;; 91 
78412 16' i! 84 
71041. 22,; 78 
68688| 29 it 71 

663371 86 J^j ^ 

11.264004; 10.000648,. J 9.999857 
61683: 50 ;^ 50 

59874 57 it 48 

570781 64 J.^1 86 



547931 
525211 



71 
78 



12j 
12 



11.250260 10.000685,., 9J»99815 

48011 92 {n 08 

45773 99 i^ 01 

43547.10.000706;; 9.999294 

413321 14 ;; 86 

39128' 21 J^ 79 

11.236935 10.000728.9 9.999272 

84754; 85;; 65 

82588' 48 ;; 57 

804221 50 ;; 50 

28273 58 ;; 42 

261341 65 ^3 85 

11.224005 10.000778.^ 9.999227 

21886 80 ;% 20 

88 il 12 

95;^ 05 

15592 10.000808 il 9.999197 

13514| 11 [l 89 

11.211446 10.000819,. 9.999181 

093871 26 , , 74 

07338 84 ;; 66 

05299 42 f; 58 

03269 50 ;% 50 

01248 58 jl 42 

11.199287 10.000866,,- 9.999184 

^' 26 

18 



Secant Cotang. 



19778 
17680 



97235 



95242! 
93258'. 
91283: 



74 



82 
90 
98 



18, 



89317! 10.000906 1^; 9.999094 



D. 



11.187369 
85411 
88471 
81589 
79616 
77702 

11^75795 
78897 
72008 
70126 
68252 
66387 

11.164529 
62679 
60887 
59002 
57175 
55856 



Tang. 



10.000914,. 9.999086 
28}* 77 

89 **! 61 

47J*; Si 

66 JT 44 

10.000964,^. 9.9990S6 



78 
81 
90 
98 
10.001007 
10.001016 



24 
88 
42 
60 
69! 



COMC 



27 
]» 
10 
02 
9.998991 
9.998984 
76 
67 
58 
60 
41 



60 

59 i 
58| 
57 I 
56 
551 
54 
53: 
52: 
,51 
50 

:49i 

47 
46 
45. 

44 

43 I 

42 

41 

40 

89 

S8 

86{ 
551 
34( 
38- 

82j 

81 

30l 

29. 

28| 
27 

«l 
25! 

24' 

2S 

22 

21 

20. 

19' 

18 

17. 

W. 
15 
14 
IS 
12' 
11 
10' 



8 

7 

6 

6; 

4: 

5! 

1 




86 Deo. 



4DBa. 



TABLB IV.— Loo. Sim, na 



SiiM 



COMG. 



Tang. 



D. 



Cotapg. I Secant 'D. 



Godne i 





1 

3 

8 

4 

5 

6 

7 

8 

9 

10 

11 

12 

IS 

14 

16 

1« 

17 

18 

19 

SO 

SI 

ss 

ss 

S4 

S6 
S6 
S7 
S8 
SO 
80 
81 
8S 
88 
84 
85 
88 
87 
88 
80 
40 
41 
48 
48 
44 
46 
48 
47 
48 
40 
60 
61 
68 
68 
64 
66 
68 
67 
68 
60 
60 



8^8586 
46887 
47188 
48971 
60761 
62626 
64291 

8.868049 
67801 
69546 
61288 
88014 
64788 

8^66465 
88166 
89888 
71585 
78255 
74988 

8^76616 
78285 
79949 
81607 
88258 
84908 

8.888542 
88174 
89801 
91421 
98086 
94848 

8.898248 
97842 
99482 

8^1017 
02598 
04189 

8.906788 
07297 
08858 
10404 
11949 
18488 

8.916082 
18660 
18078 
19591 
21108 
22810 

8.984112 
25800 
27100 
28687 
80088 
81644 

8.988015 
84481 
86942 
87898 
88850 
40298 



8005 
2992 
2980 
2987 
2956 
2948 
2981 
2919 
2908 
2898 
2884 
2878 
2881 
2850 
2889 
2828 
2817 
2808 
2795 
2788 
2778 
2768 
2752 
2742 
2781 
2721 
2711 
2700 
2890 
2880 
2870 
2660 
2851 
2841 
2881 
2622 
2812 
2808 
2598 
2584 
2575 
2586 
2558 
2547 
2588 
2529 
2520 
2512 
2508 
2494 
2486 
2477 
2488 
2460 
2452 
2448 
2485 
2427 
2419 
2411 



11.156415' 
54618 
52817 
51029 
49249 
47475 
45709 

11.148951 
42199 
40454 
88717 
88986 
85262 

11.188545 
81885 
80182 
28485 
26745 
25062 

11.128885 
21715 
20051 
18898 
16742 
15097 

11^18458 
11826, 
10199 
08579 
06965 
05857! 

11.108754> 
02158 
00568 

11.098988 
97404 
95881 

11.094264 
92708 
91147 
89596 
88051 
86512 

11.084978 
88450 
81927 
80409 
78897 
77890 

11.075888 
74891 
72900 
71418 
69982 
68456 

11.066985 
65519 
64058 
62602 
61150 
59704 



8.844644 
46455 
48260 
50057 
51846 
53628 
55408 

8.857171 
58982 
60686 
62483 
64173 
65906 

8.867632 
69851 
71064 
72770 
74469 
76162 

8.877849! 
79529! 
81202! 
82869 
84530,' 
86185 

8.887833 
89476 
91112 
92742 
94866 
95984 

8.897596 
99203 

8.900803 
02398 
03987 
05570 

8.907147 
08719 
10285 
11846 
13401 
14951 

8.916495 
18034 
19568 
21096 
22619 
24136 

8.925649 
27156 
28658 
80155 
31647 
33134 

8.984616 
36093 
37565 
89032 
40494 
41952 



3019 
8007 
2995 
2982 
2970 
2958 
2946 
2935 
2923 
2911 
2900 
2888 
2877 
2866 
2854 
2843 
2832 
2821 
2811 
2800 
2789 
2779 
2768 
2758 
2747 
2737 
2727 
2717 
2707 
2697 
2687 
2677 
2667 
2658 
2648 
2638 
2629 
2620 
2610 
2601 
2592 
2583 
2574 
2565 
2556 
2547 
2538 
2530 
2521 
2512 
2503 
2495 
2486 
2478 
2470 
2461 
2453 
2445 
2437 
2430 



11.155856 
58545 
51740 
49948 
481541 
46372; 
445971 

11.142829' 
41068 
39314, 
37567 
35827 
84094' 



10.001059,,, 
068!!! 

10.001122,,, 



140 
149 
159, 
168 



11.182368 10.001177 
80649, 
28936 
27230 
25531 ! 
23838 



I 



11.122151 
20471 
18798, 
17131 
15470 
1381 5| 

11.112167 
10524| 
088881 
07258' 
056341 
04016 

11.102404 
00797! 

11.0991971 
97602: 
96013, 
94430 

11.092853 
91281 
89715 
88154 
86509 
85049 

11.083505' 
81966 
80432 
78904 
77381 
75864 

11.074351 
72844 
. 71842 
69845 
68353 
66866 

11.065884 
63907 
62435 
60968 
59506 
58048 



187, 
196 
205 
215 
224 

10.001234 
243! 
253 
262 
272 
282 

10.001292 
301 
311 
321 
831 
841 

10.001351 

861 

371 

881 

, 891 



401 1 
10.001411: 



422,j: 

s 

452;}J 



463 
10.001473! 
484 
494 
505; 
5151 
5261 
10.001536 
547 
5581 
569 
579 
590 
10.001601 
612 
623 
634 
645 
656 



9.998941 60 
982 I 59 
923 58 
914 57 
905 ; 56 
896 55 
887 54 



58 
52 
51 
50 



9.998878 
869 
860 
851 
841 ; 49 
882 48 

9.998828 ' 47 
818 46 
804 45 
795 44 
785 . 48 
776 42 



9.998766 
757 
747 
738 
728 
718 

9.998708 
699 
689 
679 
669 
659 

9.998649 
639 
629 
619 
609 
599 

9.998589 
678 
568 
558 
548 
587 

9.998527 
516 
506 
495 
485 
474 

9.998464 
458 
442 
481 
421 
410 

9.998399 
388 
377 
866 
855 
844 



41 
40 
89 
88 
87 
86 
35 
84 
33 
32 
81 
80 
29 
28 
27 
26 
25 
24 



21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

8 

2 

1 





D. 



Secant Cotang. D. 



Tang. 



CoMG. D. 



Sine 



86 Dsa. 



u 



5 Deg. 



TABLB IV.^Loo. Snn, xtc 



hine 



I Cosoc. i Tang. ; D. i Cotang. Secant D.' Coane t 



u 


8.940-J96 


1 


41738 


2 


43174 


3 


44006 


4 


46034 


6 


47456 


6 


48874 


, 7 


8.950287 


8 


51696 


' 9 


53100 


10' 


54499 


.11' 


55S94 


112: 


57284 


13 


8.958670 


14. 


60052 


15 


61429 


16 


62801 


17 


64170 


= 18 


65534 


19 


8.966893 


20 


6S249 


21 


6y600 


.22 


70947 


23 


722S9 


24 


73628 


125 


8w974962 


'26 


7621*3 


!27 


77619 


.28 


78941 


129 


80259 


:»0 


81573- 


81 


8J82853 


32 


841S9 


33 


85491 


84 


867S9 


85 


880S3 


86 


89374 


87 


8.990660 


38 


91943 


89 


93222' 


40 


94497 


41 


95768 


42 


97036 



oirt^ 11.059704 8.94l952i 
• 58262 A«^«-» 



43 8J98299 

44 99560 

45 9.000816 
46 ; 02t>69 
47 j 03318 



48 



04563 



49 9.005805 



07044 
08278 
09510 
10737 
11962 
9.013182 
14400 
15613 
16824 
18031: 
19235 ' 
Cosine I 



2394 

2387 

2379 

2871 

2363 

2355 

2348 

2340 

2332 

2325 

2317 

2310 

2302 

2295 

22S8 

2280 

2273 

2266 

2259 

2252 

2245 

2238 

2231 

2224 

221 

2210 

2203 

2197 

2190 

2183 

2177 

2170 

2163 

2157 

2150 

2144 

2138 

2131 

2125 

2119 

2112 

2106 

2100 

2094 

2088 

2082 

2076 

2070 

2064 

2058 

2052 

2046 

2040 

2034 

2029 

2023 

2017 

2012 

2006 

IT" 



56826. 
55394 
53966 
52544 
51126 



43404 
44852 
46295 
47734 
49168 
50597 



11.049713 8.952021. 



48804 
46900 
45501 
44106 
42716 



53441 
54856 
56267 
57674 
59075 



11.041330 8.960473 



39948 
88571 
37199 
35830 
84466 
11.033107 
31751. 
30400 
29053 
27711 
26372 



6l$66 
63255 
64639 
66019 
67394 
8.968766 
70133 
71496 
72855 
74209 
75560 



^ 11.025088 8.976906 



23707 
22381 
21059, 
19741= 
18427 
11.017117 
15811, 
14509 
13211 
^ 11917. 
10626 



8248 
9586 
80921 
82251 
83577 
8.984899 
86217 
87532 
88842 
90149 
91451 



11.009340 8.992750 



0S057 
06778 
05503' 
04232 
02964 

11.001701 
00440, 

10.999184 
97931; 
96682{ 
95437i 



94045 
95337 
96624 
97908 
99188 
9.00U465 
01738 
03007 
04272 
05534 
06792 



10.994195 9.008047 



2421 
2413 
2405 
2397 
2390 
2882 
2374 
2366 
2358 
2351 
2344 
2335 
2329 
2321 
2314 
2307 
2300 
2293 
2286 
2279 
2271 
2265 
2257 
2251 
2244 
2237 
2230 
2223 
2217 
2210 
2204 
2197 
2191 
2184 
2178 
2171 
2165 
2158 
2152 
2146 
2140 
2134 
212' 



1 11.058048 

i 56596 

55146 

i 53705 

< 52266 

50882 

49403 

11.047979 

46569 

45144 

43738 

42826 

' 40925 



92956; 
91722; 
904901 
89263 
88038 



09298 
10546' 
11790; 
13031; 
14268 



10.986818 9.015502 



S5600| 
8488 
88176 
81969 
80765 
Secant 



16732 
17959 
19183 

20403 
21620 
Cotang. 



17 I 



2121 
2115 
2109 
2103 
209' 
S091 
2085 
2080 
2074 
2068 
2062 
2056 
2051 
2045 
2040 
2033 
2028 



11.089527! 
381841 
367451 
353611 
339811 
32606: 

11.081284! 
29867 
28504 
27145 
25791 
24440 
11.023094 
21752 
20414 
190791 
17749' 
16428; 
11.015101 

I 13783 

I 12468 
11158 

I 09851 
08549 

i 11.007250 
05955, 
04663 
08876 
02092 
00812 
10.999535 



10.001656 ^^ 
667 j^ 
678 jj. 
689 \l 

723 [5- 
10.001734, oi 

"^^19 

^^'19. 
<6S,q: 

791 J^i 
10.001803, A 
814 \l 
826 \l 
887 J^ 

861 20 
10.001872 ' 

884-^. 

908 *J. 
920 20| 
932 20 

10.001944 " 
956 ;0' 



968 



20; 



.r ' 



96998 

95728 

94466 

93208 

10.991958 

I 90702; 

89454 

I 88210' 

I 86969 

- 85782: 

i 10.984498 

88268 

82041; 

808171 

79697. 

78880 

Tang. 



9S0 20[ 
992 20| 

10.002004 rr 

10.002016 ^J 

10.002090 * 
103 iJi 
115 1} 
128 |i' 

153 1 I 

10.002165 on 

178 2J 

203*} 
216 2] 
2292 

10.002242 * 
255*1 
268 *i 
281 2] 
294 2} 

10.002320 poi 
838*^1 



846 
859 
872 
8861 



Cotec Id. 



9.998844 = 6k} 
333 59 
322 SS 
311 57 
300 56 
2S9 55 
277 54 

9.098266 53 
255 52 
243 51 
232 50 
220 49 , 
209 48 I 

9J»98197 47 
186 46; 
174 45' 
163 44 
151 43 
139 42 

9.998129 41 
116 40, 
104 89 
092 . 8S = 
080 87 
068 M; 

9.998056 S5 
044, 14 j 
032 S3 
020 SI. 
OOS SI 

9.997996 80 

9.997984 29 
972 ;tt 
959. r 
947 tf 
935 t5,' 
922 S4 

9.997910 S3 
897 S3' 
8S5!S1 

872;so: 

860 is' 

847113 

9.997885 1 17 1 

822'1<> 
809!15 
797,14 

784'l3; 

771, i«: 

9JM)77S8!ll, 
745 10 1 
733 9 
719 
703 
69S; 3 



9J976S0 
637 
654 
641 
6S8 
614 



84 Deo. 



95 



Deo. 



TABLE IV.— Loo. Btmn, no. 



8iae 



D. 



20485 
21682 
22825 
24016 
25208 
26886 
»jtS7M7 
28744 
29918 
81089' 
82257; 
83421 

35741! 

86896, 

880481 

89197. 

40842| 

0.0414851 

42625 

48762 

44895 

46026 

47154 

9JM8S79 

49400 

.50519 

51685 

527491 

58859 

«UIMN6' 

j 56071 

57172 

! 58271 

I 59867 

I 60460 

9.061551 



68724 
64806 
65885! 
669621 
9.0680861 
69107; 
701761 
71242 
72806, 
78866 

75480, 
76588 
77588! 
78681) 
79676 
9.080719! 
81759 
82797 
88882 
84864 
85894 



2000 
1995 
1989 
1984 
1978 
1978 
1967 
1962 
1957 
1951 
1946 
1941 
1986 
1980 
1925 
1920 
1915 
1910 
1905 
1899 
1895 
1889 
1884 
1879 
1875 
1870 
1865 
1860 
1855 
1850 
1845 
1841 
1886 
1881 
1827 
1822 
1817 
1818 
1808 
1804 
1799 
1794 
1790 
1786 
1781 
1777 

in2 

1768 
1768 
1759 
1755 
1750 
1746 
1742 
1788 
1788 
1729 
1725 
1721 
1717 



CMine 



D. 



Ootec. I Tang. 



D. 



10.980765 
79565! 
788681 
77175' 
75984 
747971 
78614; 

10i»72488 
712561 
70082: 
68911| 
67748: 
66579; 

10.965418 
64259: 
681041 
61952 
60808. 
596581 

19J58515' 
678761 



9.021620 
22834 
24044 
26251, 
26465! 
27666; 
28852| 

9.080046 
81237 
82426 
83609 
84791' 
86969 

9i)87144, 
88316, 
39486 



55105' 
58974! 
62846. 

19.951721 
60600J 
4948l{ 
488651 
47261 j 
46141> 

10.945084 
48929 
42828 
41729 
40683; 
89640 

10J88449- 
873611 
86276: 
36194 
84115 
38088 

10.981964 
80893 
29824 
28758 
27694 
26684 

10.926676 
24620 
23467 
22417 
21369 
20324 

10.919281 
18241 
17203 
16168 
16186 
14106 



40661 
41813* 
42973| 
9.044130 
46284: 
46434 
47682 
48727 



Secuit 



9^61008 
62144 
63277 
64407 
66636 
66669| 

9.067781 
68900 
60016 
61130 
62240 
63848 

9.064463 
66666 
66666 
67762 
68846 
69938! 

9.071027 
72113 
78197 
74278 
75366 
76432 

9.077606 
78676, 
79644! 
80710! 
81773! 
82833 

9.083891 
84947; 
86000 
87050 
880981 
89144| 



2023 
2017 
2011 
2006 
2000 
1995 
1990 
1986 
1979 
1974 
1969 
1964 
1968 
1958 
1948 
1943 
1938 
1933 
1928 
1923 
1918 
1913 
1908 
1903 
1898 
1693 
1889 
1884 
1879 
1874 
1870 
1866 
1860 
1856 
1861 
1846 
1842 
1837 
1833 
1828 
1824 
1819 
1816 
1810 
1806 
1802 
1797 
1793 
1789 
1784 
1780 
1776 
1772 
1767 
1763 
1769 
1765 
1751 
1747 
1743 



Cotang. 



Secant ID. I Coone 



10.97838U 

77166 

76956 

74749 

73645 

72345 

71148 

10.969954 

I 68763 

I 67675 

66391 

I 66209 

I 64031 

10.962856 

61684 

60516 

69349 

68187 

67027 
10.955870 

64716 

63666; 

624181 

51278! 
I 601311 
10.948992 10.002720 

478661 
i 46723 

46593! 
I 44465 
I 43341 



10.002386oo 

439i22 

^^^22 
466^2 

10.002480 o^ 

*^^2S 
^^^23 
53? 23 
648 23 
10.002661 „^i 

589 28 

617,23 
631 1 
ia002645 
669 



9.997614 
601 
688 
674 
661 



673 
687 
701 
716 



743 

758 
772 
786, 



10.942219 

41100 

39984! 

88870 

37760 

36662 

10.936547 

34444 

33345 

32248 

81164 

30062 

10.928973 

27887, 

268031 

26722 

24644 

23668. 

10.922495 

, 21424. 

I 20856' 

I 19290! 

I 182271 

17167| 

10.916109 

I 160631 

I 14000' 

12960 

I 11902 

I 10866 



801 
10.002816 



830 
844 
869 
873 
888 

10.002902 
917 
932 
947 
961 
976 

10.002991 

10.008006 
021 
036 
061 
066 

10.003081 
096 
111 
126 
142 
157 

10.003172 



Cotang. I D. i Tang. 



203 
218, 
284 
249 



26 



26 



Cowc. ID. 



647 66 I 
684 64 

9.997520 53 
507 52 
493 51 
480 . 50 
466 49 
452 48 

9.997489 47 
425 46 
411 i 45 
397 : 44 
383 43 
869 : 42 

9.997855 I 41 
341 I 40 
327 I 89 
313 38 
299 87 
285 , 86 

9.997271 j 35 
267 i 34 
242 '88 
228 i 82 
214 : 81 
199 180 

9.997185 29 
170 ' 28 

156 I 27 
141 > 26 

157 i 25 
112 124 

9.997098 28 
083 22 
068 I 21 
053 i 20 
039 1 19 
024 18 

9.997009 17 

9.996994 1 16 
979 ! 15 
964 14 
949 ' 13 
934 1 12 

9.996919 ! 11 
904 j 10 
889 1 9 



874 
868 
843 
9.996828 
812 
797 
782 
766 
751 



Sine 



88 Deo. 



M 



TABLE IV^-LoG. Sunn, nc. 



Cotec. 



Tang. 



Cotang. 



Secant D. Codne 



9.0&5Sik4 

1 S69'2'2 
87947 
85970 
89990 
9UXIS 
92C'24 



8 

4 

5 

6 

T 9.093"37 

8 94047 



: 9 

10 
11 
13 



95056 
96i>6'2 
97 ("^S 
9S066 



18 9.099065 

14 9.10CI062 

15 010^6 

16 0204$ 
' 17 03037 
I 13 040i5 

19 9.105010 
I 20 0599-2 



21 

23 

24 

25 

26 

27 

i2d 

;29 

,30 

31 

: 82 



9.: 



84 



135 

36 

'87 

j83 

<89i 

|40: 

421 
'48 

46 

■«i 

49 
'50 
I 51 
152 
'58 
'54 

55 

56 

57 

53 

59 

60 



06973 
07951 
03927 
09901 
110S73 
11S42 
12S09 
13774 
14737 
15693 

M 16656 
17613 
13567 
19519 
20469 
21417 

►.122362 
283U6 
24243 
25137 
26125 
27060 

L127993 
23925 
29354 
80731 
81706 
82630 

1.183551 
34470 
85387 
86303 
87216 
88123 

1.139037 
89944 
40350 
41754, 
42655| 
48555- 



1713 
17i»9 
17"4 
1700 
1696 
1692 
1^5$ 
1654 
16S0 
1676 
1673 
16d3 
16^)5 
1661 
1657 
1653 
1649 
1645 
1642 
1633 
1634 
1630 
1627 
1623 
1619 
1616 
1612 
1603 
1605 
1601 
1597 
1594 
1590 
1537 
1553 
1550 
1576 
1578 
1569 
1566 
1562 
1559 
1556 
1552 
1549 
1545 
1542 
1539 
1535 
1532 
1529 
1525 
1522 
1519 
1516 
1512 
1509 
15"6 
1503 
1500 



10.914KI6 
13073 
12053 
11030 
liHilO 
03992 
07976 

10.906963 
05953 
04944 
03933 
02935 
01934 

10.90«>935 

10.399933 
93944 
97952 
96963 
95975 

ia894990 
941*03 
93027 
92049 
91'.»73 
90099 

10^39127 
33153 
87191 
86226 
85263 
84302 

10.833344 
82337 
81433 
80431 
79531 
73533 

10.877633 
76694 
75752 
74313 
73375 
72940 

10.872007 
71075 
70146 
69219 
6S294 
67370 

10.866449 
65530 
64613 
63697 
62734 
61872' 

10.860963 

60056 

I 59150 

: 58246. 

j 57345i 

56445 



9.089144 
90187 
91223 
92266 
9330: 
94336' 
95367. 

9.096395! 
97422 
93446 
99463 

9.100437 
01504 

9.102519 
03532 
04542 
05550 
06556 
07559 

9.103560 
09559 
10556 
11551 
12543 
13533 

9.114521 
15507 
16491 
17472 
18452 
19429 

9.120404 
21877 
2284S 
23317 
242S4 
25249 

9.126211 
27172 
28130 
29037 
30041 
30994 

9.181944 
82893 
83839 
34734 
85726 
36667 

9.187605 
38542 
89476 
40409 
41340 
42269 

9.143196. 
44121 
45044, 
45966 
46S85; 
478lt3 



1738 
1735 
1731 
1727 
1722 
1719 
1715 
1711 
1707 
1703 
1699 
1695 
1691 
1637 
16S4 
1630 
1676 
1672 
1669 
1665 
1661 
1653 
1654 
1650 
1647 
1648 
1639 
1636 
1632 
1629 
1625 
1622 
1618 
1615 
1611 
1608 
1604 
16<n 
1597 
1594 
1591 
1537 
1534 
1581 
1577 
1574 
1571 
1567 
1564 
1561 
1558 
1555 
1551 
1548 
1545 
1542 
1539 
1535 
1532 
1529 



10.910856 10.008249 
09813 
087721 
07784i 
06698 
05664 
04633 



265* 
280.^ 
296 .i 



|i??? 



10.903605 
02578! 
01554' 
00532 

10.899513 
98496! 

10.89748r 
96468, 
95458 
94450 
93444 
92441 

10.891440 
90441 
89444 
83449 
87457! 
86467 

10.885479 
84498 
83509; 
82528: 
81548: 
805711 

10.879596' 
78628 
776521 
76683; 
75716 
74751' 

10.873789 
72828, 
71870 
70918 
69959! 
69006 

10.868056 
67107 
66161 
65216 
64274 
63333- 

10.862895 
61458' 
60524 
595911 
58660' 
57781 

10.856804 
55879; 
54956 
54034 
53116 
5219' 



10.008359 Q« 

488 51 
10.003454 c^ 
470;: 
486 2^ 
^2 27 

635 $i 
10.008551 L, 
667 £ 
583 |ii 
000 Jli 
616 21 
63251 
10.003649 ' 
665 5I' 

682$;: 

698 5^1 
716 ||. 
jSlgg 

10.008748 „ft 
765^1 

4 81 Qoi 

798 1. 
815 H 

I0.003S49 Z. 
866,5, 
88S|°I 
900^1 
917 H 
934 » 

10.003951 Z: 
968^ 

10.004002 *r 

10.004054 Zl\ 
072» 
080 ?» 
106^ 

141 ** 

"29 

10.004159^ 

i94,|; 

212;S 

229» 
24'r» 



9.996751 . 60 
735 59 
720 '58 
704 57 
688 56 
673:65 
657 54 

9.996641 53 
625 53 
610 ' 51 
594 50 
578 49 
562 48 

9.996546 47 
580 4«, 
514 45: 
498 44 
482 43 
465 42 

9.996449 41 1 
438 40; 
417 59' 
400 3Si 
884 37 
86S S6: 

9.996851 35 
335 31! 
818 33! 
802 32i 
285 31 ! 
269 30 

9.996253 29 
235 28i 
219 27 
202 26 
185 25, 
168 24 

9.996151 23 1 
134 22i 
117 21 
100 20 
033 19 
066 18 

9.996049 17 
032 10 
015 15 

9.995998 14 
980! 13 j 
968 12! 

9.996946 ill 
928:10 



911 




894 




876 




859 




9.995841 




833 




806 




788 




771 




753 




Sne 


r 



i Coaine | D. , gwamt Cotang. | D. 



Tang. I Coiec D 



82 Dec. 



27 



8 Deo. 



TABLE IV.— Loo. Sihbs, xto. 



Sine 



D. 



Coicc. Tang. 



Cotang. Secant D.| Cosine 




1 
2 
8 
4 
6 
6 
7 
8 
» 
10 
11 
12 
18 
14 
16 
18 
17 
18 
19 
20 
21 
22 
28 
24 
25 
26 
27 
28 
20 
80 
81 
82 
88 
84 
86 
88 
87 
88 
80 
40 
41 
42 
48 
44 
46 
48 
47 
48 
49 
60 
61 
62 
68 
64 
66 
5$ 
67 
68 
69 
80 



9.148655 
4453 
5849 
6243 
7186 
8026 
8915 
9.149802 
9.150686 
1569 
2451 
8330 
4208 
9.166088 
6957 
6830 
7700 
8569 
9435 
9.160801 
1164 
2026 
2885 
8743 
4600 
9a66454 
6807 
7159 
8008 
8856 
9702 
9.170647 
1889 
2230 
8070 
^ 8908 
4744 
9.176678 
6411 
7242 
8072 
8900 
9726 
9.180651 
1874 
2196 
8016 
8884 
4651 
9.186466 
6280 
7092 
7908 
8712 
9519 
9.190825 
1180 
1983 
2784 
8584 
4832 

Godne 



1496 

1493 

1490 

1487 

1484 

1481 

1478 

1476 

1472 

1469 

1466 

1463 

1460 

1457 

1454 

1451 

1448 

1445 

1442 

1439 

1436 

1433 

1430 

1427 

1424 

1422 

1419 

1416 

1413 

1410 

1407 

1405 

1402 

1899 

1896 

1894 

1391 

1388 

1886 

1883 

1380 

1377 

1874 

1372 

1869 

1866 

1364 

1861 

1859 

1356 

1353 

1851 

1348 

1346 

1843 

1341 

1338 

1336 

1338 

1880 



10.856445 

5547 

4651 

3757 

2864 

1974 

1085 

10.850198 

10.849814 

8431 

7549 

6670 

5792 

10.844917 

4043 

3170 

2300 

1431 

0565 

10.839699 

8836 

7975 

7116 

6257 

5400 

10.834546 

3693 

2841 

1992 

1144 

0298 

10.829458 

8611 

7770 

6930 

6092 

5256 

10.824422 

3589 

2758 

1928 

1100 

0274 

10.819449 

8626 

7804 

6984 

6166 

5349 

10.814534 

3720 

2908 

2097 

1288 

0481 

10.809675 

8870 

8067 

7266 

6466 

566 8 

Secant 



9.147808 
8718 
9632 

9.150544 
1454 
2363 
8269 

9.154174 
5077 
5978 
6877 
7776 
8671 

9.159565 

9.160457 
1347 



3128 
4008 

9.164892 
5774 
6654 
7682 
8409 
9284 

9.170167 
1029 
1899 
2767 
3634 
4499 

9.175362 
6224 
7084 
7942 
8799 
9655 

9.180508 
1360 
2211 
8059 
8907 
4752 

9.185597 
6489 
7280 
8120 
8958 
9794 

9.190629 
1462 
2294 
3124 
3953 
4780 

9.195606 
6430 
7253 
8074 
8894 
9718 
Cotang. 



1526 

1523 

1520 

1617 

1514 

1611 

1508 

1505 

1502 

1499 

1496 

1493 

1490 

1487 

1484 

1481 

1479 

1476 

1473 

1470 

1467 

1464 

1461 

1458 

1455 

1453 

1450 

1447 

1444 

1442 

1439 

1436 

1433 

1431 

1428 

1425 

1423 

1420 

1417 

1415 

1412 

1409 

1407 

1404 

1402 

1399 

1396 

1393 

1391 

1389 

1886 

1884 

1381 

1379 

1376 

1374 

1371 

1869 

1366 

1364 



10.852197 
1282 
0368 

10.849456 
8546 
7687 
6781 

10.845826 
4923 
4022 
8123 
2225 
1329 

10.840435 

10.839543 
8653 
7764 
6877 
5992 

10.885108 
4226 
8846 
2468 
1591 
0716 



10.004247 
265 
283 
301 
319 
336 
854 

10.004372 
390 
409, 
427 



4451' 
468 

10.004481 
499 
518 
536 
554 
573 

10.004591 
610 
628 
647 

eee 

684 



10.829843 10.004703 



8971 
8101 
7283 
6366 
5501 



722 

740 
759 
778 
797 



10.824688! 10.004816 
37761 886 
2916 854 



2058 
1201 
0345 

10.819492 
8640 
7789 
6941 
6093 
5248 

10.814403 
3561 
2720 
1880 
1042 
0206 

10.809371 
8538 
7706 
6876 
6047 
5220 

10.804394 
3570 
2747 
1926 
1106 

0287